Question:
Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered $1$ through $10$ in some order. Each block in layers $2, 3$ and $4$ is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
<image1>

Answer: 114

solid geometry

Question:
Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered $1$ through $10$ in some order. Each block in layers $2, 3$ and $4$ is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
<image1>

Answer: 114


Question:
In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle.  $ CO$ is drawn and extended to $ D$. $ AO$ is drawn.  Which of the following expresses the relationship between $ x$ and $ y$?

<image1>

Choices:
- $x=3y \\$
- $x=2y \\$
- $x=60^\circ \\$
- $\text{there is no special relationship between }x\text{ and }y \\$
- $x=2y \text{ or }x=3y\text{, depending upon the length of }AB$

metric geometry - angle

Question:
In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle.  $ CO$ is drawn and extended to $ D$. $ AO$ is drawn.  Which of the following expresses the relationship between $ x$ and $ y$?

<image1>
(A) $x=3y \\$
(B) $x=2y \\$
(C) $x=60^\circ \\$
(D) $\text{there is no special relationship between }x\text{ and }y \\$
(E) $x=2y \text{ or }x=3y\text{, depending upon the length of }AB$


Question:
Inside a unit square a certain area has been coloured in black. In which square is the black area biggest?
<image1>

Choices:
- A
- B
- C
- D
- E

metric geometry - area

Question:
Inside a unit square a certain area has been coloured in black. In which square is the black area biggest?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: In the diagram, if $\triangle ABC$ and $\triangle PQR$ are equilateral, then what is the measure of $\angle CXY$ in degrees? <image1>

Answer: 40

Solution:
Since $\triangle ABC$ and $\triangle PQR$ are equilateral, then $\angle ABC=\angle ACB=\angle RPQ=60^\circ$.

Therefore, $\angle YBP = 180^\circ-65^\circ-60^\circ=55^\circ$ and $\angle YPB = 180^\circ-75^\circ-60^\circ=45^\circ$.

In $\triangle BYP$, we have $\angle BYP = 180^\circ - \angle YBP - \angle YPB = 180^\circ - 55^\circ-45^\circ=80^\circ$.

Since $\angle XYC = \angle BYP$, then $\angle XYC=80^\circ$.

In $\triangle CXY$, we have $\angle CXY = 180^\circ - 60^\circ - 80^\circ = 40^\circ$.

So our final answer is $\boxed{40}$ degrees.

metric geometry - angle

Question: In the diagram, if $\triangle ABC$ and $\triangle PQR$ are equilateral, then what is the measure of $\angle CXY$ in degrees? <image1>

Answer: 40


Question:
In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is:
<image1>

Choices:
- $r\sqrt{2}$
- $r$
- $\text{not a side of an inscribed regular polygon}$
- $\frac{r\sqrt{3}}{2}$
- $r\sqrt{3}$

metric geometry - length

Question:
In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is:
<image1>
(A) $r\sqrt{2}$
(B) $r$
(C) $\text{not a side of an inscribed regular polygon}$
(D) $\frac{r\sqrt{3}}{2}$
(E) $r\sqrt{3}$


Question:
A regular octagon is folded exactly in half three times until a triangle is obtained. The bottom corner of the triangle is then cut off with a cut perpendicular to one side of the triangle as shown.
<image1>
Which of the following will be seen when the triangle is unfolded?
<image2>

Choices:
- A
- B
- C
- D
- E

Solution: Let the angles $x, y$ and $z$ be as shown in the diagram. Since the triangle containing angle $x$ at its vertex is formed by folding the octagon in half three times, $x=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times 360^{\circ}=45^{\circ}$. Therefore, since angles in a triangle add to $180^{\circ}$ and the cut is made at $90^{\circ}$ to the side of that triangle, $y$ is also $45^{\circ}$. Therefore, since the dotted line is a line along which the quadrilateral shown was folded to create the triangle, it is a  ![](https://cdn.mathpix.com/cropped/2023_12_27_23f752c0584f83753c92g-06.jpg?height=305&width=317&top_left_y=884&top_left_x=1552) line of symmetry for the quadrilateral and hence $z=y=45^{\circ}$. Hence the angle at the corner of the hole when the octagon is unfolded will be $2 \times 45^{\circ}=90^{\circ}$ and therefore the hole will be in the shape of a square with its upper side horizontal, as is shown in diagram C.

transformation geometry

Question:
A regular octagon is folded exactly in half three times until a triangle is obtained. The bottom corner of the triangle is then cut off with a cut perpendicular to one side of the triangle as shown.
<image1>
Which of the following will be seen when the triangle is unfolded?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$.

<image1>

Choices:
- $32$
- $16 + 16\sqrt{3}$
- $48$
- $32 + 16\sqrt{3}$
- $64$

metric geometry - area

Question:
Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$.

<image1>
(A) $32$
(B) $16 + 16\sqrt{3}$
(C) $48$
(D) $32 + 16\sqrt{3}$
(E) $64$


Question:
A road leads away from each of the six houses (see diagram). A hexagon showing the roads in the middle is however, missing. Which hexagons fit in the middle so
<image1>that one can travel from $A$ to $B$ and to $E$, but not to $D$?

Choices:
- 1 and 2
- 1 and 4
- 1 and 5
- 2 and 3
- 4 and 5

graph theory

Question:
A road leads away from each of the six houses (see diagram). A hexagon showing the roads in the middle is however, missing. Which hexagons fit in the middle so
<image1>that one can travel from $A$ to $B$ and to $E$, but not to $D$?
(A) 1 and 2
(B) 1 and 4
(C) 1 and 5
(D) 2 and 3
(E) 4 and 5


Question:
Chantal has placed numbers in two of the nine cells (see diagram). She wants to place the numbers 1, 2, 3 in every row and every column exactly once. How big is the sum of the two numbers in the grey cells?
<image1>

Answer: 4

logic

Question:
Chantal has placed numbers in two of the nine cells (see diagram). She wants to place the numbers 1, 2, 3 in every row and every column exactly once. How big is the sum of the two numbers in the grey cells?
<image1>

Answer: 4


Question:
Gray and white pearls are threaded onto a string. Tony pulls pearls from the ends of the chain. After pulling off the fifth gray pearl he stops. At most, how many white pearls could he have pulled off?
<image1>

Answer: 7

arithmetic

Question:
Gray and white pearls are threaded onto a string. Tony pulls pearls from the ends of the chain. After pulling off the fifth gray pearl he stops. At most, how many white pearls could he have pulled off?
<image1>

Answer: 7


Question:
Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is
<image1>

Choices:
- $\frac{1}{6}$
- $\frac{1}{4}$
- $\frac{1}{3}$
- $\frac{2}{5}$
- $\frac{1}{2}$

metric geometry - area

Question:
Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is
<image1>
(A) $\frac{1}{6}$
(B) $\frac{1}{4}$
(C) $\frac{1}{3}$
(D) $\frac{2}{5}$
(E) $\frac{1}{2}$


Question: In the diagram shown here (which is not drawn to scale), suppose that $\triangle ABC \sim \triangle PAQ$ and $\triangle ABQ \sim \triangle QCP$.  If $m\angle BAC = 70^\circ$, then compute $m\angle PQC$. <image1>

Answer: 15

Solution: We're given that $\triangle ABQ \sim \triangle QCP$ and thus $m\angle B = m\angle C.$ Therefore, $\triangle ABC$ is isosceles. From the given $m\angle BAC=70^\circ$, we have that $m\angle ABC = m\angle BCA = 55^\circ$.  But we also know that $\triangle ABC \sim \triangle PAQ$, which means that $m\angle PAQ=55^\circ$ as well.  Subtracting, $m\angle BAQ=15^\circ$.  Finally, from similar triangles, we have $m\angle PQC=m\angle BAQ = \boxed{15^\circ}$.

metric geometry - angle

Question: In the diagram shown here (which is not drawn to scale), suppose that $\triangle ABC \sim \triangle PAQ$ and $\triangle ABQ \sim \triangle QCP$.  If $m\angle BAC = 70^\circ$, then compute $m\angle PQC$. <image1>

Answer: 15


Question:
<image1>

In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees?

Answer: 15

metric geometry - angle

Question:
<image1>

In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees?

Answer: 15


Question:
In a rectangle we draw both diagonals and the segment which joins a vertex with the midpoint of one of the sides, as shown in the picture. What is the result of dividing the length of the diagonal by the length of segment $O P$?
<image1>

Answer: 6

metric geometry - length

Question:
In a rectangle we draw both diagonals and the segment which joins a vertex with the midpoint of one of the sides, as shown in the picture. What is the result of dividing the length of the diagonal by the length of segment $O P$?
<image1>

Answer: 6


Question:
There are three flowers on the back of the left cactus. In total, the cactus on the right has six more flowers than the cactus on the left. How many flowers are on the back of the right cactus?
<image1>

Answer: 12

arithmetic

Question:
There are three flowers on the back of the left cactus. In total, the cactus on the right has six more flowers than the cactus on the left. How many flowers are on the back of the right cactus?
<image1>

Answer: 12


Question:
Two concentric circles with radii 1 and 9 form an annulus. $n$ circles without overlap are drawn inside this annulus, where every circle touches both circles of the annulus. (The diagram shows an example for $\mathrm{n}=1$ and the other radii as given.) What is the biggest possible value of $n$?
<image1>

Answer: 3

combinatorial geometry

Question:
Two concentric circles with radii 1 and 9 form an annulus. $n$ circles without overlap are drawn inside this annulus, where every circle touches both circles of the annulus. (The diagram shows an example for $\mathrm{n}=1$ and the other radii as given.) What is the biggest possible value of $n$?
<image1>

Answer: 3


Question:
Anna made the figure on the right out of five cubes. Which of the following figures (when seen from any direction) cannot she get from the figure on the right side if she is allowed to move exactly one cube?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

combinatorial geometry

Question:
Anna made the figure on the right out of five cubes. Which of the following figures (when seen from any direction) cannot she get from the figure on the right side if she is allowed to move exactly one cube?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
This piece of paper was folded in half twice, and then had two equilateral triangles cut out of it. Which diagram shows how the paper will look when it is unfolded again? <image1>
<image2>

Choices:
- A
- B
- C
- D
- E

Solution: The diagram shows how the cut-out triangles form rhombic holes when the paper is unfolded. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-009.jpg?height=212&width=209&top_left_y=508&top_left_x=1529)

transformation geometry

Question:
This piece of paper was folded in half twice, and then had two equilateral triangles cut out of it. Which diagram shows how the paper will look when it is unfolded again? <image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The diagram shows four congruent right-angled triangles inside a rectangle. What is the total area, in $\mathrm{cm}^{2}$, of the four triangles? <image1>

Answer: 56

Solution: Let the lengths of the two shorter sides of the right-angled triangles be $a \mathrm{~cm}$ and $b \mathrm{~cm}$ as shown in the diagram. From the diagram in the question, it can be seen that $2 b=28$ and $a+2 b=30$, and hence $b=14$ and $a=2$. Therefore the total area, in $\mathrm{cm}^{2}$, of the four triangles is $4 \times\left(\frac{1}{2} \times 2 \times 14\right)=56$.

metric geometry - area

Question: The diagram shows four congruent right-angled triangles inside a rectangle. What is the total area, in $\mathrm{cm}^{2}$, of the four triangles? <image1>

Answer: 56


Question:
Which of the figures shown bellow cannot be cut out of the figure illustrated nearby?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

combinatorial geometry

Question:
Which of the figures shown bellow cannot be cut out of the figure illustrated nearby?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: In the diagram, $P R S V$ is a rectangle with $P R=20 \mathrm{~cm}$ and $P V=12 \mathrm{~cm}$. Jeffrey marks points $U$ and $T$ on $V S$ and $Q$ on $P R$ as shown. What is the shaded area? <image1>

Choices:
- More information needed
- $60 \mathrm{~cm}^{2}$
- $100 \mathrm{~cm}^{2}$
- $110 \mathrm{~cm}^{2}$
- $120 \mathrm{~cm}^{2}$

Solution: Consider the two unshaded triangles. Each has height equal to $12 \mathrm{~cm}$ and hence their total area is $\left(\frac{1}{2} \times P Q \times 12+\frac{1}{2} \times Q R \times 12\right) \mathrm{cm}^{2}=6 \times(P Q+Q R) \mathrm{cm}^{2}=6 \times 20 \mathrm{~cm}^{2}=120 \mathrm{~cm}^{2}$. Therefore the shaded area is $(20 \times 12-120) \mathrm{cm}^{2}=120 \mathrm{~cm}^{2}$.

metric geometry - area

Question:
In the diagram, $P R S V$ is a rectangle with $P R=20 \mathrm{~cm}$ and $P V=12 \mathrm{~cm}$. Jeffrey marks points $U$ and $T$ on $V S$ and $Q$ on $P R$ as shown. What is the shaded area? <image1>
(A) More information needed
(B) $60 \mathrm{~cm}^{2}$
(C) $100 \mathrm{~cm}^{2}$
(D) $110 \mathrm{~cm}^{2}$
(E) $120 \mathrm{~cm}^{2}$


Question:
Paul folds a piece of paper, then punches a hole into the paper and unfolds it again. The unfolded paper then looks like the picture on the right. Along which dotted line can Paul have folded the paper beforehand?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

transformation geometry

Question:
Paul folds a piece of paper, then punches a hole into the paper and unfolds it again. The unfolded paper then looks like the picture on the right. Along which dotted line can Paul have folded the paper beforehand?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Mr. Hide wants to find a treasure that he buried years before. He can only remember that he buried the treasure at least $5 \mathrm{~m}$ away from the hedge and no more than $5 \mathrm{~m}$ away from the old pear tree. Which picture shows best the area where Mr. Hide has to look for the treasure?
<image1>

Choices:
- A
- B
- C
- D
- E

metric geometry - area

Question:
Mr. Hide wants to find a treasure that he buried years before. He can only remember that he buried the treasure at least $5 \mathrm{~m}$ away from the hedge and no more than $5 \mathrm{~m}$ away from the old pear tree. Which picture shows best the area where Mr. Hide has to look for the treasure?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Six different numbers, chosen from integers 1 to 9 , are written on the faces of a cube, one number per face. The sum of the numbers on each pair of opposite faces is always the same. Which of the following numbers could have been written on the opposite side with the number 8 ?
<image1>

Answer: 3

logic

Question:
Six different numbers, chosen from integers 1 to 9 , are written on the faces of a cube, one number per face. The sum of the numbers on each pair of opposite faces is always the same. Which of the following numbers could have been written on the opposite side with the number 8 ?
<image1>

Answer: 3


Question:
<image1>

Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined.  How many of the resulting figures can be folded into a topless cubical box?

Answer: 6

solid geometry

Question:
<image1>

Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined.  How many of the resulting figures can be folded into a topless cubical box?

Answer: 6


Question:
In the diagram on the right, $Q S R$ is a straight line, $\angle Q P S=12^{\circ}$ and $P Q=P S=R S$.
What is the size of $\angle Q P R$ ? <image1>

Choices:
- $36^{\circ}$
- $42^{\circ}$
- $54^{\circ}$
- $60^{\circ}$
- $84^{\circ}$

Solution: Observing that triangle $P Q S$ is isosceles, we have $\angle P S Q=\frac{1}{2}\left(180^{\circ}-12^{\circ}\right)=84^{\circ}$ and hence $\angle P S R=180^{\circ}-84^{\circ}=96^{\circ}$.  Since triangle $P R S$ is also isosceles, we have $\angle S P R=\frac{1}{2}\left(180^{\circ}-96^{\circ}\right)=42^{\circ}$.  Hence $\angle Q P R=12^{\circ}+42^{\circ}=54^{\circ}$.

metric geometry - angle

Question:
In the diagram on the right, $Q S R$ is a straight line, $\angle Q P S=12^{\circ}$ and $P Q=P S=R S$.
What is the size of $\angle Q P R$ ? <image1>
(A) $36^{\circ}$
(B) $42^{\circ}$
(C) $54^{\circ}$
(D) $60^{\circ}$
(E) $84^{\circ}$


Question:
Five blocks are built with equal cubes glued face to face. In which of them was the smallest number of cubes used?
<image1>

Choices:
- A
- B
- C
- D
- E

solid geometry

Question:
Five blocks are built with equal cubes glued face to face. In which of them was the smallest number of cubes used?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Leon has drawn a closed path on the surface of a cuboid. Which net can represent his path?
<image1>

Choices:
- A
- B
- C
- D
- E

descriptive geometry

Question:
Leon has drawn a closed path on the surface of a cuboid. Which net can represent his path?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In this diagram a scheme is indicated for associating all the points of segment $ \overline{AB}$ with those of segment $ \overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \overline{AB}$ to $ D$ and let $ y$ be the distance from the associated point $ P'$ of $ \overline{A'B'}$ to $ D'$. Then for any pair of associated points, if $ x = a,\, x + y$ equals:
<image1>

Choices:
- $13a$
- $17a - 51$
- $17 - 3a$
- $\frac{17 - 3a}{4}$
- $12a - 34$

metric geometry - length

Question:
In this diagram a scheme is indicated for associating all the points of segment $ \overline{AB}$ with those of segment $ \overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \overline{AB}$ to $ D$ and let $ y$ be the distance from the associated point $ P'$ of $ \overline{A'B'}$ to $ D'$. Then for any pair of associated points, if $ x = a,\, x + y$ equals:
<image1>
(A) $13a$
(B) $17a - 51$
(C) $17 - 3a$
(D) $\frac{17 - 3a}{4}$
(E) $12a - 34$


Question:
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School.  The total number of students at East is $2000$ and at West, $2500$.  In the two schools combined, the percent of students who prefer tennis is

<image1>

Choices:
- $30\%$
- $31\%$
- $32\%$
- $33\%$
- $34\%$

statistics

Question:
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School.  The total number of students at East is $2000$ and at West, $2500$.  In the two schools combined, the percent of students who prefer tennis is

<image1>
(A) $30\%$
(B) $31\%$
(C) $32\%$
(D) $33\%$
(E) $34\%$


Question:
What is the piece that fits completely to the given one to form a rectangle?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

combinatorial geometry

Question:
What is the piece that fits completely to the given one to form a rectangle?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The following four statements, and only these are found on a card:
<image1>
(Assume each statement is either true or false.) Among them the number of false statements is exactly

Answer: 3

logic

Question:
The following four statements, and only these are found on a card:
<image1>
(Assume each statement is either true or false.) Among them the number of false statements is exactly

Answer: 3


Question:
Let $ABCD$ be a rectangle.  Let $E$ and $F$ be points on $BC$ and $CD$, respectively, so that the areas of triangles $ABE$, $ADF$, and $CEF$ are 8, 5, and 9, respectively.  Find the area of rectangle $ABCD$.

<image1>

Answer: 40

Solution:
Let $u = BE$, $v = CE$, $x = CF$, and $y = DF$. [asy]
unitsize(1.5 cm);

pair A, B, C, D, E, F;

A = (0,2);
B = (0,0);
C = (3,0);
D = (3,2);
E = (3*B + 2*C)/5;
F = (2*D + C)/3;

draw(A--B--C--D--cycle);
draw(A--E--F--cycle);

label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, NE);
label("$E$", E, S);
label("$F$", F, dir(0));
label("$8$", (A + B + E)/3);
label("$5$", (A + D + F)/3);
label("$9$", (C + E + F)/3);

label("$u$", (B + E)/2, S);
label("$v$", (C + E)/2, S);
label("$x$", (C + F)/2, dir(0));
label("$y$", (D + F)/2, dir(0));
[/asy] Then the area of triangle $ABE$ is $u(x + y)/2 = 8$, so $u(x + y) = 16$.  The area of triangle $ADF$ is $y(u + v)/2 = 5$, so $y(u + v) = 10$.  The area of triangle $CEF$ is $xv/2 = 9$, so $xv = 18$.  Thus, we have the system of equations \begin{align*}
ux + uy &= 16, \\
uy + vy &= 10, \\
vx &= 18.
\end{align*} Solving for $x$ in equation (1), we find \[x = \frac{16 - uy}{u}.\] Solving for $v$ in equation (2), we find \[v = \frac{10 - uy}{y}.\] Substituting into equation (3), we get \[\frac{10 - uy}{y} \cdot \frac{16 - uy}{u} = 18.\] This equation simplifies to \[u^2 y^2 - 44uy + 160 = 0.\] We recognize this equation as a quadratic in $uy$, which factors as $(uy - 4)(uy - 40) = 0$.  From equation (1), $uy$ must be less than 16, so $uy = 4$.

Then from equation (1), $ux = 16 - uy = 16 - 4 = 12$, and from equation (2), $vy = 10 - uy = 10 - 4 = 6$.  Therefore, the area of rectangle $ABCD$ is $(u + v)(x + y) = ux + uy + vx + vy = 12 + 4 + 18 + 6 = \boxed{40}$.

metric geometry - area

Question:
Let $ABCD$ be a rectangle.  Let $E$ and $F$ be points on $BC$ and $CD$, respectively, so that the areas of triangles $ABE$, $ADF$, and $CEF$ are 8, 5, and 9, respectively.  Find the area of rectangle $ABCD$.

<image1>

Answer: 40


Question:
Beatrice has a few grey tiles that all look exactly like the one pictured. At least how many of these tiles does she need in order to make a complete square?
<image1>

Answer: 4

arithmetic

Question:
Beatrice has a few grey tiles that all look exactly like the one pictured. At least how many of these tiles does she need in order to make a complete square?
<image1>

Answer: 4


Question: Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$?\n<image1>

Answer: $\frac{1}{2}+\frac{\sqrt{3}}{3}$

Solution: We easily find $E F=1$, so since $\angle B=60^{\circ}$, we know $B F=\frac{1}{\sqrt{3}}$, so $B C=1+\frac{1}{\sqrt{3}}$. \n, the area of $A B C$ is\n$$\n\frac{\sqrt{3}}{4}\left(1+\frac{1}{\sqrt{3}}\right)^2=\frac{1}{2}+\frac{\sqrt{3}}{3} .\n$$

metric geometry - area

Question: Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$?\n<image1>

Answer: $\frac{1}{2}+\frac{\sqrt{3}}{3}$


Question: A large square is divided into smaller squares, as shown. What fraction of the large square is shaded grey? <image1>

Choices:
- $\frac{2}{3}$
- $\frac{2}{5}$
- $\frac{4}{7}$
- $\frac{4}{9}$
- $\frac{5}{12}$

Solution: The largest grey square is a quarter of the large square. The smaller grey squares are each one ninth of the size of the largest grey square. Hence the fraction of the large square which is shaded is $\frac{1}{4}+\frac{7}{9} \times \frac{1}{4}=\frac{1}{4} \times\left(1+\frac{7}{9}\right)=\frac{1}{4} \times \frac{16}{9}=\frac{4}{9}$

combinatorial geometry

Question:
A large square is divided into smaller squares, as shown. What fraction of the large square is shaded grey? <image1>
(A) $\frac{2}{3}$
(B) $\frac{2}{5}$
(C) $\frac{4}{7}$
(D) $\frac{4}{9}$
(E) $\frac{5}{12}$


Question: Costa is building a new fence in his garden. He uses 25 planks of wood, each of which is $30 \mathrm{~cm}$ long. He arranges these planks so that there is the same slight overlap between any two adjacent planks, as shown in the diagram. The total length of Costa's new fence is 6.9 metres. What is the length in centimetres of the overlap between any pair of adjacent planks? <image1>

Answer: 2.5

Solution: Let the length of the overlap be $y \mathrm{~cm}$. From the diagram in the question, it can be seen that the total length of the fence can be calculated as the total length of the 13 pieces in the lower row in the diagram and the total length of the 12 planks in the upper row of the diagram, with each of the 12 having two overlaps removed. Hence $690=13 \times 30+12 \times(30-2 y)$. Therefore $690=390+360-24 y$ and hence $24 y=60$. This has solution $y=2.5$. Therefore the overlap between adjacent planks is $2.5 \mathrm{~cm}$.

metric geometry - length

Question: Costa is building a new fence in his garden. He uses 25 planks of wood, each of which is $30 \mathrm{~cm}$ long. He arranges these planks so that there is the same slight overlap between any two adjacent planks, as shown in the diagram. The total length of Costa's new fence is 6.9 metres. What is the length in centimetres of the overlap between any pair of adjacent planks? <image1>

Answer: 2.5


Question:
In the picture the large square has an area of 1. What is the area of the small black square?
<image1>

Choices:
- $\frac{1}{100}$
- $\frac{1}{300}$
- $\frac{1}{600}$
- $\frac{1}{900}$
- $\frac{1}{1000}$

arithmetic

Question:
In the picture the large square has an area of 1. What is the area of the small black square?
<image1>
(A) $\frac{1}{100}$
(B) $\frac{1}{300}$
(C) $\frac{1}{600}$
(D) $\frac{1}{900}$
(E) $\frac{1}{1000}$


Question:
The faces of the prism shown, are made up of two triangles and three squares. The six vertices are labelled using the numbers 1 to 6. The sum of the four numbers around each square is always the same. The numbers 1 and 5 are given in the diagram. Which number is written at vertex $X$?
<image1>

Answer: 2

algebra

Question:
The faces of the prism shown, are made up of two triangles and three squares. The six vertices are labelled using the numbers 1 to 6. The sum of the four numbers around each square is always the same. The numbers 1 and 5 are given in the diagram. Which number is written at vertex $X$?
<image1>

Answer: 2


Question:
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?

<image1>

Answer: 10

combinatorial geometry

Question:
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?

<image1>

Answer: 10


Question:
In this picture there is what I saw on four different clocks at the same time. Only one of them had the right time. One was 20 minutes fast. Another 20 minutes slow. One had stopped some time ago.
<image1>
What was the right time?

Choices:
- 4:45
- 5:05
- 5:25
- 5:40
- 12:00

logic

Question:
In this picture there is what I saw on four different clocks at the same time. Only one of them had the right time. One was 20 minutes fast. Another 20 minutes slow. One had stopped some time ago.
<image1>
What was the right time?
(A) 4:45
(B) 5:05
(C) 5:25
(D) 5:40
(E) 12:00


Question:
Six different digits from the set
\[\{ 1,2,3,4,5,6,7,8,9\}\]
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.
The sum of the six digits used is

<image1>

Answer: 29

combinatorics

Question:
Six different digits from the set
\[\{ 1,2,3,4,5,6,7,8,9\}\]
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.
The sum of the six digits used is

<image1>

Answer: 29


Question:
Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of the roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?
<image1>

Choices:
- Amy
- Ben
- Carol
- David
- It cannot be determined from the drawing.

logic

Question:
Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of the roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?
<image1>
(A) Amy
(B) Ben
(C) Carol
(D) David
(E) It cannot be determined from the drawing.


Question:
Which of the following diagrams shows the locus of the midpoint of the wheel when the wheel rolls along the zig-zag curve shown?
<image1>

Choices:
- A
- B
- C
- D
- E

Solution: As the wheel goes over the top it pivots around the peak so the midpoint travels through a circular arc. At the troughs the wheel changes directions in an instant from down-right to up-right, so the midpoint travels through a sharp change of direction. This gives the locus in diagram $\mathrm{E}$.

transformation geometry

Question:
Which of the following diagrams shows the locus of the midpoint of the wheel when the wheel rolls along the zig-zag curve shown?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A road leads away from each of the six houses (see diagram). A hexagon showing the roads in the middle is however, missing. Which hexagons fit in the middle so
<image1>that one can travel from $A$ to $B$ and to $E$, but not to $D$?

Choices:
- 1 and 2
- 1 and 4
- 1 and 5
- 2 and 3
- 4 and 5

graph theory

Question:
A road leads away from each of the six houses (see diagram). A hexagon showing the roads in the middle is however, missing. Which hexagons fit in the middle so
<image1>that one can travel from $A$ to $B$ and to $E$, but not to $D$?
(A) 1 and 2
(B) 1 and 4
(C) 1 and 5
(D) 2 and 3
(E) 4 and 5


Question: A square has two of its vertices on a semicircle and the other two on the diameter of the semicircle as shown. The radius of the circle is 1 . What is the area of the square? <image1>

Choices:
- $\frac{4}{5}$
- $\frac{\pi}{4}$
- 1
- $\frac{4}{3}$
- $\frac{2}{\sqrt{3}}$

Solution: Let $O$ be the centre of the circle, and $P, Q, P^{\prime}$ and $Q^{\prime}$ the vertices of the square. The triangles $O P Q$ and $O P^{\prime} Q^{\prime}$ are congruent since they are right-angled and have two equal sides $\left(P Q=P^{\prime} Q^{\prime}\right.$ since they are edges of a square, and $O P=O P^{\prime}$ because each is a radius). Hence $O Q=O Q^{\prime}$, and $O$ is thus the midpoint of the edge of the square.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-268.jpg?height=302&width=443&top_left_y=109&top_left_x=1526)  Let the side of the square be $2 x$. And $O P=1$ since it is a radius. And by Pythagoras' Theorem on triangle $O P Q$ we have $(2 x)^{2}+x^{2}=1^{2}$, so $4 x^{2}+x^{2}=1$, and $5 x^{2}=1$. This gives $x^{2}=\frac{1}{5}$. The area of the square is $(2 x)^{2}=4 x^{2}=4 \times \frac{1}{5}=\frac{4}{5}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-268.jpg?height=300&width=443&top_left_y=478&top_left_x=1526)

metric geometry - area

Question:
A square has two of its vertices on a semicircle and the other two on the diameter of the semicircle as shown. The radius of the circle is 1 . What is the area of the square? <image1>
(A) $\frac{4}{5}$
(B) $\frac{\pi}{4}$
(C) 1
(D) $\frac{4}{3}$
(E) $\frac{2}{\sqrt{3}}$


Question:
A square floor is tiled with congruent square tiles.  The tiles on the two diagonals of the floor are black.  The rest of the tiles are white.  If there are 101 black tiles, then the total number of tiles is

<image1>

Answer: 2601

algebra

Question:
A square floor is tiled with congruent square tiles.  The tiles on the two diagonals of the floor are black.  The rest of the tiles are white.  If there are 101 black tiles, then the total number of tiles is

<image1>

Answer: 2601


Question:
Four identical rhombuses (diamonds) and two squares are fitted together to form a regular octagon as shown. How big are the obtuse interior angles in the rhombuses?
<image1>

Choices:
- $135^{\circ}$
- $140^{\circ}$
- $144^{\circ}$
- $145^{\circ}$
- $150^{\circ}$

solid geometry

Question:
Four identical rhombuses (diamonds) and two squares are fitted together to form a regular octagon as shown. How big are the obtuse interior angles in the rhombuses?
<image1>
(A) $135^{\circ}$
(B) $140^{\circ}$
(C) $144^{\circ}$
(D) $145^{\circ}$
(E) $150^{\circ}$


Question: In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right? <image1>

Answer: 158

Solution: The numbers on the left use the digits $\mathrm{M}$ and $\mathrm{K}$ in the Units column, and the digits $\mathrm{J}$ and $\mathrm{L}$ in the Tens column. The numbers on the right use the digits $\mathrm{M}$ and $\mathrm{K}$ twice in the Units columns, and the digits $\mathrm{J}$ and $\mathrm{L}$ twice in the Tens column, so the total is exactly twice that of the numbers on the left. Twice 79 is 158 , answer B.

algebra

Question: In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right? <image1>

Answer: 158


Question: Let there be a unit square initially tiled with four congruent shaded equilateral triangles, as seen below. The total area of all of the shaded regions can be expressed in the form $\frac{a-b\sqrt{c}}{d}$ , where $a, b, c$, and $d$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c + d$.\n<image1>

Answer: 14

metric geometry - area

Question: Let there be a unit square initially tiled with four congruent shaded equilateral triangles, as seen below. The total area of all of the shaded regions can be expressed in the form $\frac{a-b\sqrt{c}}{d}$ , where $a, b, c$, and $d$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c + d$.\n<image1>

Answer: 14


Question:
Clown Pipo looks like this:
<image1>
He looks at himself in the mirror. Which picture does he see?
<image2>

Choices:
- A
- B
- C
- D
- E

descriptive geometry

Question:
Clown Pipo looks like this:
<image1>
He looks at himself in the mirror. Which picture does he see?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Two rectangles $A B C D$ and $D B E F$ are shown in the figure. What is the area (in $\mathrm{cm}^{2}$ ) of the rectangle $D B E F$?
<image1>

Answer: 12

metric geometry - area

Question:
Two rectangles $A B C D$ and $D B E F$ are shown in the figure. What is the area (in $\mathrm{cm}^{2}$ ) of the rectangle $D B E F$?
<image1>

Answer: 12


Question:
The big rectangle consists of various squares of different sizes. Each of the three smallest squares has area 1. How big is the area of the big rectangle?
<image1>

Answer: 77

combinatorial geometry

Question:
The big rectangle consists of various squares of different sizes. Each of the three smallest squares has area 1. How big is the area of the big rectangle?
<image1>

Answer: 77


Question: The target shown consists of an inner black circle with two rings, one black and one white, around it. The width of each ring is equal to the radius of the black circle. What is the ratio of the area of the black ring to the area of the inner black circle? <image1>

Choices:
- $2: 1$
- $3: 1$
- $4: 1$
- $5: 1$
- $6: 1$

Solution: Let the radius of the inner circle be $R$ so its area is then $\pi R^{2}$. The white ring and the inner black circle together have area $\pi(2 R)^{2}=4 \pi R^{2}$. The whole target has area $\pi(3 R)^{2}=9 \pi R^{2}$. Hence the outer black circle has area $9 \pi R^{2}-4 \pi R^{2}=5 \pi R^{2}$ which is 5 times the area of the inner circle.

metric geometry - area

Question:
The target shown consists of an inner black circle with two rings, one black and one white, around it. The width of each ring is equal to the radius of the black circle. What is the ratio of the area of the black ring to the area of the inner black circle? <image1>
(A) $2: 1$
(B) $3: 1$
(C) $4: 1$
(D) $5: 1$
(E) $6: 1$


Question:
Barbara wants to complete the grid shown on the right by inserting three numbers into the empty spaces. The sum of the first three numbers should be 100 , the sum of the middle three numbers 200 and the sum of the last three numbers 300. Which is the middle number in this grid?
<image1>

Answer: 60

algebra

Question:
Barbara wants to complete the grid shown on the right by inserting three numbers into the empty spaces. The sum of the first three numbers should be 100 , the sum of the middle three numbers 200 and the sum of the last three numbers 300. Which is the middle number in this grid?
<image1>

Answer: 60


Question:
Which shape has the biggest area?
<image1>

Choices:
- A
- B
- C
- D
- E

counting

Question:
Which shape has the biggest area?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) = 6$ have?
<image1>

Answer: 6

analytic geometry

Question:
The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) = 6$ have?
<image1>

Answer: 6


Question:
Tom has these nine cards:
<image1>
He places these cards on the board next to each other so that each horizontal line and each vertical line has three cards with the three different shapes and the three different amounts of drawings. He has already placed three cards, as shown in the picture. Which card should he place in the colored box?
<image2>
<image3>

Choices:
- A
- B
- C
- D
- E

logic

Question:
Tom has these nine cards:
<image1>
He places these cards on the board next to each other so that each horizontal line and each vertical line has three cards with the three different shapes and the three different amounts of drawings. He has already placed three cards, as shown in the picture. Which card should he place in the colored box?
<image2>
<image3>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: In the diagram the large square is divided into 25 smaller squares. Adding up the sizes of the five angles $X P Y, X Q Y, X R Y, X S Y$ and $X T Y$, what total is obtained? <image1>

Choices:
- $30^{\circ}$
- $45^{\circ}$
- $60^{\circ}$
- $75^{\circ}$
- $90^{\circ}$

Solution: The diagram shows the angles rearranged to form a $45^{\circ}$ angle. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-011.jpg?height=348&width=756&top_left_y=1088&top_left_x=1024)

combinatorial geometry

Question:
In the diagram the large square is divided into 25 smaller squares. Adding up the sizes of the five angles $X P Y, X Q Y, X R Y, X S Y$ and $X T Y$, what total is obtained? <image1>
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\circ}$
(D) $75^{\circ}$
(E) $90^{\circ}$


Question:
The rectangle $A B C D$ is made up of 12 congruent rectangles (see diagram). How big is the ratio $\frac{A D}{D C}$?
<image1>

Choices:
- $\frac{8}{9}$
- $\frac{5}{6}$
- $\frac{7}{8}$
- $\frac{2}{3}$
- $\frac{9}{8}$

combinatorial geometry

Question:
The rectangle $A B C D$ is made up of 12 congruent rectangles (see diagram). How big is the ratio $\frac{A D}{D C}$?
<image1>
(A) $\frac{8}{9}$
(B) $\frac{5}{6}$
(C) $\frac{7}{8}$
(D) $\frac{2}{3}$
(E) $\frac{9}{8}$


Question:
In $ \triangle ABC$, we have $ \angle C = 3 \angle A$, $ a = 27$, and $ c = 48$.  What is $ b$?

<image1>

Choices:
- $33$
- $35$
- $37$
- $39$
- $\text{not uniquely determined}$

metric geometry - length

Question:
In $ \triangle ABC$, we have $ \angle C = 3 \angle A$, $ a = 27$, and $ c = 48$.  What is $ b$?

<image1>
(A) $33$
(B) $35$
(C) $37$
(D) $39$
(E) $\text{not uniquely determined}$


Question:
Andrea has wound some rope around a piece of wood, as shown in the diagram on the right. She rotates the wood $180^{\circ}$ as shown by the arrow in the diagram. What does she see after the rotation?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

Solution: When Andrea started, the rope passed through each notch at the top of the piece of wood. When the wood is rotated through $180^{\circ}$, the rope must now pass through each notch at the bottom of the piece of wood. This means, of the options available, she must see D.  However, there is an alternative view that she could see after the rotation. Consider the initial piece of wood and label the notches along the top 1,2,3,4 and the notches along the bottom $P, Q, R, S$ as shown in the diagram. The reverse of the piece of wood could also show $P$ to 3 , then $S$ to 2 and finally $R$ to 4 .  1234  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-114.jpg?height=270&width=342&top_left_y=280&top_left_x=1505)  $P Q R S$

transformation geometry

Question:
Andrea has wound some rope around a piece of wood, as shown in the diagram on the right. She rotates the wood $180^{\circ}$ as shown by the arrow in the diagram. What does she see after the rotation?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9$. What is the area of $ABCD?$

<image1>

Answer: 240

metric geometry - area

Question:
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9$. What is the area of $ABCD?$

<image1>

Answer: 240


Question:
The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?

<image1>

Choices:
- $\text{2 o'clock}$
- $\text{3 o'clock}$
- $\text{4 o'clock}$
- $\text{6 o'clock}$
- $\text{8 o'clock}$

transformation geometry

Question:
The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?

<image1>
(A) $\text{2 o'clock}$
(B) $\text{3 o'clock}$
(C) $\text{4 o'clock}$
(D) $\text{6 o'clock}$
(E) $\text{8 o'clock}$


Question: The diagram shows a square, an equilateral triangle and a regular pentagon. What is the size of $\angle W U V$ ? <image1>

Choices:
- $21^{\circ}$
- $23^{\circ}$
- $25^{\circ}$
- $27^{\circ}$
- $29^{\circ}$

Solution: The interior angles of an equilateral triangle, a square and a regular pentagon are $180^{\circ} \div 3=60^{\circ}$, $2 \times 180^{\circ} \div 4=90^{\circ}$ and $3 \times 180^{\circ} \div 5=108^{\circ}$ respectively. Therefore the size of the obtuse $\angle U V W$ is $108^{\circ}+90^{\circ}-60^{\circ}=138^{\circ}$. Since the pentagon and the square share a side and the square and the equilateral triangle also share a side, the side-length of the pentagon is equal to the side-length of the equilateral triangle. Therefore $U V=V W$ and hence the triangle $U V W$ is isosceles and $\angle V W U=\angle W U V$. Therefore the size of $\angle W U V$ is $\left(180^{\circ}-138^{\circ}\right) \div 2=21^{\circ}$.

metric geometry - angle

Question:
The diagram shows a square, an equilateral triangle and a regular pentagon. What is the size of $\angle W U V$ ? <image1>
(A) $21^{\circ}$
(B) $23^{\circ}$
(C) $25^{\circ}$
(D) $27^{\circ}$
(E) $29^{\circ}$


Question:
A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$?

<image1>

Choices:
- 1:2
- 3:5
- 2:3
- 3:4
- 4:5

transformation geometry

Question:
A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$?

<image1>
(A) 1:2
(B) 3:5
(C) 2:3
(D) 3:4
(E) 4:5


Question: The solid shown was formed by cutting a right circular cylinder in half. If the base has a radius of 6 cm and the height is 10 cm, what is the total surface area, in terms of $\pi$, of the solid? <image1>

Answer: 96\pi+120

Solution:
The surface area of the solid can be split into four pieces:  the top and bottom, the curved side, and the flat side.

The top and bottom pieces are semicircles with radius 6; together, they add up to a circle with radius 6 and area $\pi(6^2)=36\pi$.

The curved side can be rolled out to a rectangle with height 10.  The width of this rectangle is half the circumference of the base of the cylinder, which is $\frac{1}{2}\cdot 2\pi\cdot 6 = 6\pi$.  Thus the area of the curved side is $10\cdot 6\pi = 60\pi$.

The flat side is a rectangle with height 10.  The width of this rectangle is the diameter of the cylinder, which is $6\cdot 2 = 12$.  Thus the area of the flat side is $10\cdot 12 = 120$.

Finally, the total surface area of the solid is $36\pi+60\pi+120=\boxed{96\pi+120}$.

solid geometry

Question: The solid shown was formed by cutting a right circular cylinder in half. If the base has a radius of 6 cm and the height is 10 cm, what is the total surface area, in terms of $\pi$, of the solid? <image1>

Answer: 96\pi+120


Question:
In the diagram you see the rectangular garden of Green's family. It has an area of $30 \mathrm{~m}^{2}$ and is divided into three rectangular parts. One side of the part where flowers are growing has a length of $2 \mathrm{~m}$. Its area is $10 \mathrm{~m}^{2}$. The part with strawberries has one side of length $3 \mathrm{~m}$. What is the area of the part where vegetables are growing?
<image1>

Choices:
- $4 \mathrm{~m}^{2}$
- $6 \mathrm{~m}^{2}$
- $8 \mathrm{~m}^{2}$
- $10 \mathrm{~m}^{2}$
- $12 \mathrm{~m}^{2}$

metric geometry - area

Question:
In the diagram you see the rectangular garden of Green's family. It has an area of $30 \mathrm{~m}^{2}$ and is divided into three rectangular parts. One side of the part where flowers are growing has a length of $2 \mathrm{~m}$. Its area is $10 \mathrm{~m}^{2}$. The part with strawberries has one side of length $3 \mathrm{~m}$. What is the area of the part where vegetables are growing?
<image1>
(A) $4 \mathrm{~m}^{2}$
(B) $6 \mathrm{~m}^{2}$
(C) $8 \mathrm{~m}^{2}$
(D) $10 \mathrm{~m}^{2}$
(E) $12 \mathrm{~m}^{2}$


Question: Barbara wants to place draughts on a $4 \times 4$ board in such a way that the number of draughts in each row and in each column are all different (she may place more than one draught in a square, and a square may be empty). What is the smallest number of draughts that she would need? <image1>

Answer: 14

Solution: There are four rows and four columns, so we need eight different sums. The smallest eight sums (if possible) would be $0,1,2,3, \ldots, 7$. Since each draught is counted towards the sum of a row and the sum of a column, we would need $\frac{1}{2}(0+1+2+\ldots+7)=14$ draughts. The diagram shows it is possible to place 14 draughts on the board to create the eight smallest sums (the numbers in the cells represent how many  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-107.jpg?height=254&width=257&top_left_y=1852&top_left_x=1571) draughts there are in each cell, and the column and row totals are shown).

combinatorics

Question: Barbara wants to place draughts on a $4 \times 4$ board in such a way that the number of draughts in each row and in each column are all different (she may place more than one draught in a square, and a square may be empty). What is the smallest number of draughts that she would need? <image1>

Answer: 14


Question:
<image1>

Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is

Choices:
- $4\sqrt{5}$
- $\sqrt{65}$
- $2\sqrt{17}$
- $3\sqrt{7}$
- $6\sqrt{2}$

metric geometry - length

Question:
<image1>

Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is
(A) $4\sqrt{5}$
(B) $\sqrt{65}$
(C) $2\sqrt{17}$
(D) $3\sqrt{7}$
(E) $6\sqrt{2}$


Question:
The plane is tiled by congruent squares and congruent pentagons as indicated.
<image1>
The percent of the plane that is enclosed by the pentagons is closest to

Choices:
- 50
- 52
- 54
- 56
- 58

metric geometry - area

Question:
The plane is tiled by congruent squares and congruent pentagons as indicated.
<image1>
The percent of the plane that is enclosed by the pentagons is closest to
(A) 50
(B) 52
(C) 54
(D) 56
(E) 58


Question:
Now it is 2008. What is the total sum of these digits?
<image1>

Answer: 10

arithmetic

Question:
Now it is 2008. What is the total sum of these digits?
<image1>

Answer: 10


Question:
$A B C D$ is a square with side length $10 \mathrm{~cm}$. The distance of $\mathrm{N}$ to $\mathrm{M}$ measures $6 \mathrm{~cm}$. Each area not shaded grey is either a sqaure or an isosceles triangle. How big is the area shaded in grey?
<image1>

Choices:
- $42 \mathrm{~cm}^{2}$
- $46 \mathrm{~cm}^{2}$
- $48 \mathrm{~cm}^{2}$
- $52 \mathrm{~cm}^{2}$
- $58 \mathrm{~cm}^{2}$

combinatorial geometry

Question:
$A B C D$ is a square with side length $10 \mathrm{~cm}$. The distance of $\mathrm{N}$ to $\mathrm{M}$ measures $6 \mathrm{~cm}$. Each area not shaded grey is either a sqaure or an isosceles triangle. How big is the area shaded in grey?
<image1>
(A) $42 \mathrm{~cm}^{2}$
(B) $46 \mathrm{~cm}^{2}$
(C) $48 \mathrm{~cm}^{2}$
(D) $52 \mathrm{~cm}^{2}$
(E) $58 \mathrm{~cm}^{2}$


Question:
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?

<image1>

Answer: 26

algebra

Question:
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?

<image1>

Answer: 26


Question:
Four of the following five diagrams can be drawn without lifting the pencil and without going over a line twice. For one diagram this is not true. Which one is it?
<image1>

Choices:
- A
- B
- C
- D
- E

graph theory

Question:
Four of the following five diagrams can be drawn without lifting the pencil and without going over a line twice. For one diagram this is not true. Which one is it?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Four identical dice were put together to make a tower as shown. The sum of the numbers on opposite faces of each dice is always 7. What would the tower look like from behind?
<image1>
<image2>

Choices:
- A)
- B)
- C)
- D)
- E)

descriptive geometry

Question:
Four identical dice were put together to make a tower as shown. The sum of the numbers on opposite faces of each dice is always 7. What would the tower look like from behind?
<image1>
<image2>
(A) A)
(B) B)
(C) C)
(D) D)
(E) E)


Question:
The parabola in the figure has an equation of the form $y=a x^{2}+b x+c$ for some distinct real numbers $a, b$ and $c$. Which of the following equations could be an equation of the line in the figure?
<image1>

Choices:
- $y=b x+c$
- $y=c x+b$
- $y=a x+b$
- $y=a x+c$
- $y=c x+a$

analytic geometry

Question:
The parabola in the figure has an equation of the form $y=a x^{2}+b x+c$ for some distinct real numbers $a, b$ and $c$. Which of the following equations could be an equation of the line in the figure?
<image1>
(A) $y=b x+c$
(B) $y=c x+b$
(C) $y=a x+b$
(D) $y=a x+c$
(E) $y=c x+a$


Question:
In the picture you can see a number flower. Mary pulled out all the leaves with numbers which give remainder 2 when divided by 6. What is the sum of the numbers on the leaves that Mary pulled out?
<image1>

Answer: 46

algebra

Question:
In the picture you can see a number flower. Mary pulled out all the leaves with numbers which give remainder 2 when divided by 6. What is the sum of the numbers on the leaves that Mary pulled out?
<image1>

Answer: 46


Question:
The star shown in the picture is made by fitting together 12 congruent equilateral triangles. The perimeter of the star is $36 \mathrm{~cm}$. What is the perimeter of the grey hexagon?
<image1>

Choices:
- $6 \mathrm{~cm}$
- $12 \mathrm{~cm}$
- $18 \mathrm{~cm}$
- $24 \mathrm{~cm}$
- $30 \mathrm{~cm}$

combinatorial geometry

Question:
The star shown in the picture is made by fitting together 12 congruent equilateral triangles. The perimeter of the star is $36 \mathrm{~cm}$. What is the perimeter of the grey hexagon?
<image1>
(A) $6 \mathrm{~cm}$
(B) $12 \mathrm{~cm}$
(C) $18 \mathrm{~cm}$
(D) $24 \mathrm{~cm}$
(E) $30 \mathrm{~cm}$


Question:
A digital clock shows the following time: <image1> What time is it when it uses the exactly same digits again for the first time after that?
<image2>

Choices:
- A
- B
- C
- D
- E

combinatorics

Question:
A digital clock shows the following time: <image1> What time is it when it uses the exactly same digits again for the first time after that?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A cuboid with surface area $X$ is cut up along six planes parallel to the sides (see diagram). What is the total surface area of all 27 thus created solids?
<image1>

Choices:
- $X$
- $2 X$
- $3 X$
- $4 X$
- It depends on the position of the planes.

solid geometry

Question:
A cuboid with surface area $X$ is cut up along six planes parallel to the sides (see diagram). What is the total surface area of all 27 thus created solids?
<image1>
(A) $X$
(B) $2 X$
(C) $3 X$
(D) $4 X$
(E) It depends on the position of the planes.


Question:
In the picture on the right a number should be written next to each point. The sum of the numbers on the corners of each side of the hexagon should be equal. Two numbers have already been written. Which number should be in the place marked ' $x$ '?
<image1>

Answer: 1

logic

Question:
In the picture on the right a number should be written next to each point. The sum of the numbers on the corners of each side of the hexagon should be equal. Two numbers have already been written. Which number should be in the place marked ' $x$ '?
<image1>

Answer: 1


Question: Let $T$ be the answer from the previous part. Rectangle $R$ has length $T$ times its width. $R$ is inscribed in a square $S$ such that the diagonals of $ S$ are parallel to the sides of $R$. What proportion of the area of $S$ is contained within $R$?\n<image1>

Answer: $\frac{8}{25}$

Solution: Let the width of rectangle $R$ be $2x.$ Then, the length of rectangle $R$ is $8x.$\n\nIt follows that the side length of square $S$ is$$\sqrt{2}x+4\sqrt{2}x=5\sqrt{2}x,$$so the area of the square is$$50x^2.$$\nHence, the answer is$$\frac{16x^2}{50x^2} = \boxed{\frac{8}{25}}.$$

metric geometry - area

Question: Let $T$ be the answer from the previous part. Rectangle $R$ has length $T$ times its width. $R$ is inscribed in a square $S$ such that the diagonals of $ S$ are parallel to the sides of $R$. What proportion of the area of $S$ is contained within $R$?\n<image1>

Answer: $\frac{8}{25}$


Question:
Five clocks are hanging on the wall. One clock is one hour ahead. Another one is one hour late and one is correct. Two clocks have stopped working. Which clock shows the correct time?
<image1>

Choices:
- A
- B
- C
- D
- E

arithmetic

Question:
Five clocks are hanging on the wall. One clock is one hour ahead. Another one is one hour late and one is correct. Two clocks have stopped working. Which clock shows the correct time?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: When the bat Elisa left its cave at night, the digital clock showed <image1>. When she came back in the morning and hung herself upside down, she looked at her watch and saw <image1>. How long did she stay out of the cave?

Choices:
- 2h 48m
- 2h 59m
- 3h 39m
- 3h 41m
- 3h 49m

arithmetic

Question:
When the bat Elisa left its cave at night, the digital clock showed <image1>. When she came back in the morning and hung herself upside down, she looked at her watch and saw <image1>. How long did she stay out of the cave?
(A) 2h 48m
(B) 2h 59m
(C) 3h 39m
(D) 3h 41m
(E) 3h 49m


Question:
In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?
<image1>

Choices:
- $\frac{3}{4}$
- $\frac{3}{2}$
- $2$
- $\frac{12}{5}$
- $\frac{5}{2}$

metric geometry - area

Question:
In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?
<image1>
(A) $\frac{3}{4}$
(B) $\frac{3}{2}$
(C) $2$
(D) $\frac{12}{5}$
(E) $\frac{5}{2}$


Question: Blahaj has two rays with a common endpoint A0 that form an angle of $1^o$. They construct a sequence of points $A_0$, $. . . $, $A_n$ such that for all $1 \le i \le n$, $|A_{i-1}A_i | = 1$, and $|A_iA_0| > |A_{i-1}A_0|$. Find the largest possible value of $n$.\n<image1>

Answer: 90

Solution: Starting with the second, each new line segment encloses a new isosceles triangle. A simple angle chase yields successive base angles of $1^{\circ}, 2^{\circ}, 3^{\circ}$, and so on in each of these triangles. We can go as far as a base angle of $89^{\circ}$, as trying to go further will cause us to repeat points, which does not increase distance from $A_0$. The 89th triangle appears when the 90 th segment does, meaning the maximal $n$ is 90 .

transformation geometry

Question: Blahaj has two rays with a common endpoint A0 that form an angle of $1^o$. They construct a sequence of points $A_0$, $. . . $, $A_n$ such that for all $1 \le i \le n$, $|A_{i-1}A_i | = 1$, and $|A_iA_0| > |A_{i-1}A_0|$. Find the largest possible value of $n$.\n<image1>

Answer: 90


Question:
The digits of the sequence $123451234512345 \ldots$ fill the cells on a sheet of paper in a spiral-like manner beginning with the marked cell (see the figure). Which digit is written in the cell being 100 cells above the marked one?
<image1>

Answer: 1

algebra

Question:
The digits of the sequence $123451234512345 \ldots$ fill the cells on a sheet of paper in a spiral-like manner beginning with the marked cell (see the figure). Which digit is written in the cell being 100 cells above the marked one?
<image1>

Answer: 1


Question:
Which quadrant contains no points of the graph of the linear function $f(x)=-3.5 x+7$?
<image1>

Choices:
- I
- II
- III
- IV
- Every quadrant contains at least one point of the graph.

analytic geometry

Question:
Which quadrant contains no points of the graph of the linear function $f(x)=-3.5 x+7$?
<image1>
(A) I
(B) II
(C) III
(D) IV
(E) Every quadrant contains at least one point of the graph.


Question:
The statements on the right give clues to the identity of a four-digit number.
<image1>
What is the last digit of the four-digit number?

Answer: 3

Solution: Let's call the four-digit number $N$. The last clue tells us that none of the digits 7, 6, 4 or 2 is a digit in $N$. Then the fourth clue shows that 1 is a digit in $N$, but it is not the fourth digit. The first clue now tells us that $N$ involves a 3 but not as its third digit. It also shows that 1 is not the second digit. The second clue now tells us that either 8 is the second digit of $N$ and 9 is not one of its digits or else 9 is the first digit of $N$ and 8 is not one of its digits. Suppose that 8 were the correct second digit. Then the third clue would tell us that both 0 and 5 were correct digits. But this would mean that all of 1, 3, 8, 0 and 5 were digits of the four-digit number $N$. Therefore 8 is incorrect and so 9 is correct and is the first digit. Knowing this, the third clue shows us that exactly one of 5 and 0 is correct and, moreover, it is in the right place. It can't be 5 because the first digit of $N$ is 9 . So 0 is the correct second digit. We already know 3 is correct, but is not the third digit; so the last digit of $N$ is 3 and $N$ is 9013 .

logic

Question:
The statements on the right give clues to the identity of a four-digit number.
<image1>
What is the last digit of the four-digit number?

Answer: 3


Question:
A $1 \times 1 \times 1$ cube is cut out of each corner of a $3 \times 3 \times 3$ cube. The picture shows the result after the first cube is cut out. How many faces will the final shape have?
<image1>

Answer: 30

solid geometry

Question:
A $1 \times 1 \times 1$ cube is cut out of each corner of a $3 \times 3 \times 3$ cube. The picture shows the result after the first cube is cut out. How many faces will the final shape have?
<image1>

Answer: 30


Question:
The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:
<image1>

Choices:
- 4: 3
- 3: 2
- 4: 1
- 3: 1
- 6: 1

metric geometry - length

Question:
The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:
<image1>
(A) 4: 3
(B) 3: 2
(C) 4: 1
(D) 3: 1
(E) 6: 1


Question:
The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

<image1>

Choices:
- $\text{The mean increases by 1 and the median does not change.}$
- $\text{The mean increases by 1 and the median increases by 1.}$
- $\text{The mean increases by 1 and the median increases by 5.}$
- $\text{The mean increases by 5 and the median increases by 1.}$
- $\text{The mean increases by 5 and the median increases by 5.}$

statistics

Question:
The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

<image1>
(A) $\text{The mean increases by 1 and the median does not change.}$
(B) $\text{The mean increases by 1 and the median increases by 1.}$
(C) $\text{The mean increases by 1 and the median increases by 5.}$
(D) $\text{The mean increases by 5 and the median increases by 1.}$
(E) $\text{The mean increases by 5 and the median increases by 5.}$


Question:
Zilda took a square sheet of paper of side 1 and made two folds taking two consecutive sides of the sheet to a diagonal of it, as shown in the picture, obtaining a quadrilateral (highlighted outline). What is the area of this quadrilateral?
<image1>

Choices:
- $\frac{7}{10}$
- $2-\sqrt{2}$
- $\frac{3}{5}$
- $\sqrt{2}-1$
- $\frac{\sqrt{2}}{2}$

transformation geometry

Question:
Zilda took a square sheet of paper of side 1 and made two folds taking two consecutive sides of the sheet to a diagonal of it, as shown in the picture, obtaining a quadrilateral (highlighted outline). What is the area of this quadrilateral?
<image1>
(A) $\frac{7}{10}$
(B) $2-\sqrt{2}$
(C) $\frac{3}{5}$
(D) $\sqrt{2}-1$
(E) $\frac{\sqrt{2}}{2}$


Question:
Six points are placed and numbered as shown on the right. Two triangles are drawn: one by connecting the even numbered points, and one by connecting the odd numbered points. Which of the following shapes is the result?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

combinatorial geometry

Question:
Six points are placed and numbered as shown on the right. Two triangles are drawn: one by connecting the even numbered points, and one by connecting the odd numbered points. Which of the following shapes is the result?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
How much does this Ferris wheel need to turn so that a white gondola is on top for
the first time?
<image1>

Choices:
- $\frac{1}{2}$ turn
- $\frac{1}{3}$ turn
- $\frac{1}{6}$ turn
- $\frac{1}{12}$ turn
- $\frac{5}{6}$ turn

arithmetic

Question:
How much does this Ferris wheel need to turn so that a white gondola is on top for
the first time?
<image1>
(A) $\frac{1}{2}$ turn
(B) $\frac{1}{3}$ turn
(C) $\frac{1}{6}$ turn
(D) $\frac{1}{12}$ turn
(E) $\frac{5}{6}$ turn


Question: A solid $5\times 5\times 5$ cube is composed of unit cubes. Each face of the large, solid cube is partially painted with gray paint, as shown. <image1> 	 	What fraction of the entire solid cube's unit cubes have no paint on them?

Answer: \frac{69}{125}

Solution: We know that each of the unit cubes in the $3\times3\times3$ cube in the center of the $5\times5\times5$ cube has no paint on it. On the surface of the cube, three of the unit cubes on each edge of the big cube have no paint on them, and the center unit cube of each face of the big cube has no paint on it. Since a cube has $12$ edges and $6$ faces, this makes a total of $3\cdot3\cdot3 + 12\cdot3 + 6\cdot1 = 69$ unit cubes with no paint on them. There are $125$ unit cubes altogether. The fraction with no paint is $\boxed{\frac{69}{125}}.$

combinatorial geometry

Question: A solid $5\times 5\times 5$ cube is composed of unit cubes. Each face of the large, solid cube is partially painted with gray paint, as shown. <image1> 	 	What fraction of the entire solid cube's unit cubes have no paint on them?

Answer: \frac{69}{125}


Question:
Mr Hyde can't remember exactly where he has hidden his treasure. He knows it is at least $5 \mathrm{~m}$ from his hedge, and at most $5 \mathrm{~m}$ from his tree. Which of the following shaded areas could represent the largest region where his treasure could lie?
<image1>

Choices:
- A
- B
- C
- D
- E

Solution: Points which are at most $5 \mathrm{~m}$ from the tree lie on or inside a circle of radius $5 \mathrm{~m}$ with its centre at the tree. However, not all of the inside of the circle will be shaded because the treasure is at least $5 \mathrm{~m}$ from the hedge, so we should have an unshaded rectangular strip next to the hedge. This leaves the shaded region in A.

metric geometry - area

Question:
Mr Hyde can't remember exactly where he has hidden his treasure. He knows it is at least $5 \mathrm{~m}$ from his hedge, and at most $5 \mathrm{~m}$ from his tree. Which of the following shaded areas could represent the largest region where his treasure could lie?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The diagram shows a spiral of consecutive numbers starting with 1. In which order will the numbers 625, 626 and 627 appear in the spiral? <image1>

Choices:
- $625 \uparrow 626 \uparrow 627$
- $625 \uparrow 626 \rightarrow$
- $625 \rightarrow 626 \rightarrow 627$
- $625 \rightarrow 626 \uparrow 627$
- $625 \downarrow 626 \downarrow 627$

algebra

Question:
The diagram shows a spiral of consecutive numbers starting with 1. In which order will the numbers 625, 626 and 627 appear in the spiral? <image1>
(A) $625 \uparrow 626 \uparrow 627$
(B) $625 \uparrow 626 \rightarrow$
(C) $625 \rightarrow 626 \rightarrow 627$
(D) $625 \rightarrow 626 \uparrow 627$
(E) $625 \downarrow 626 \downarrow 627$


Question:
18 cubes are coloured white or grey or black and are arranged as shown.
<image1>
The figures below show the white and the black parts. Which of the following is the grey part?
<image2>
<image3>

Choices:
- (A)
- (B)
- (C)
- (D)
- (E)

combinatorial geometry

Question:
18 cubes are coloured white or grey or black and are arranged as shown.
<image1>
The figures below show the white and the black parts. Which of the following is the grey part?
<image2>
<image3>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
Where is the Kangaroo?
<image1>

Choices:
- In the circle and in the triangle but not in the square.
- In the circle and in the square but not in the triangle.
- In the triangle and in the square but not in the circle.
- In the circle but in neither the square or the triangle.
- In the square but in neither the circle or the triangle.

combinatorial geometry

Question:
Where is the Kangaroo?
<image1>
(A) In the circle and in the triangle but not in the square.
(B) In the circle and in the square but not in the triangle.
(C) In the triangle and in the square but not in the circle.
(D) In the circle but in neither the square or the triangle.
(E) In the square but in neither the circle or the triangle.


Question:
If I join the midpoints of the sides of the large triangle in the picture, a small triangle is formed. If I join the midpoints of the sides of this small triangle, a tiny triangle is formed. How many of these tiny triangles can fit into the largest triangle at the same time?
<image1>

Answer: 16

metric geometry - area

Question:
If I join the midpoints of the sides of the large triangle in the picture, a small triangle is formed. If I join the midpoints of the sides of this small triangle, a tiny triangle is formed. How many of these tiny triangles can fit into the largest triangle at the same time?
<image1>

Answer: 16


Question:
In the overlapping triangles $ \triangle{ABC}$ and $ \triangle{ABE}$ sharing common side $ AB$, $ \angle{EAB}$ and $ \angle{ABC}$ are right angles, $ AB = 4$, $ BC = 6$, $ AE = 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle{ADE}$ and $ \triangle{BDC}$?

<image1>

Answer: 4

metric geometry - area

Question:
In the overlapping triangles $ \triangle{ABC}$ and $ \triangle{ABE}$ sharing common side $ AB$, $ \angle{EAB}$ and $ \angle{ABC}$ are right angles, $ AB = 4$, $ BC = 6$, $ AE = 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle{ADE}$ and $ \triangle{BDC}$?

<image1>

Answer: 4


Question:
Six cars are parked on a parking. Someone wants to move from $S$ to $F$. His route must be as short as possible. Which of the following routes is the shortest?
<image1>

Choices:
- A
- B
- C
- D
- All routes are equal

combinatorial geometry

Question:
Six cars are parked on a parking. Someone wants to move from $S$ to $F$. His route must be as short as possible. Which of the following routes is the shortest?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) All routes are equal


Question:
Every one of these six building blocks consists of 5 little cubes. The little cubes are either white or grey. Cubes of equal colour don't touch each other. How many little white cubes are there in total?
<image1>

Answer: 12

solid geometry

Question:
Every one of these six building blocks consists of 5 little cubes. The little cubes are either white or grey. Cubes of equal colour don't touch each other. How many little white cubes are there in total?
<image1>

Answer: 12


Question:
Toninho wants to write strictly positive and consecutive whole numbers, in the nine places of the figure, so that the sum of the three numbers in each diameter is equal to 24. What is the largest possible sum for all the nine numbers?
<image1>

Answer: 81

logic

Question:
Toninho wants to write strictly positive and consecutive whole numbers, in the nine places of the figure, so that the sum of the three numbers in each diameter is equal to 24. What is the largest possible sum for all the nine numbers?
<image1>

Answer: 81


Question:
Which of the figures is shown most often in the sequence?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

counting

Question:
Which of the figures is shown most often in the sequence?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Each of the four balls weighs either 10 or 20 or 30 or 40 grams. Which ball weighs 30 grams?
<image1>

Choices:
- A
- B
- C
- D
- It can be A or B.

logic

Question:
Each of the four balls weighs either 10 or 20 or 30 or 40 grams. Which ball weighs 30 grams?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) It can be A or B.


Question:
In the accompanying figure $ \overline{CE}$ and $ \overline{DE}$ are equal chords of a circle with center $ O$. Arc $ AB$ is a quarter-circle. Then the ratio of the area of triangle $ CED$ to the area of triangle $ AOB$ is:
<image1>

Choices:
- $\sqrt{2} : 1$
- $\sqrt{3} : 1$
- $4 : 1$
- $3 : 1$
- $2 : 1$

metric geometry - area

Question:
In the accompanying figure $ \overline{CE}$ and $ \overline{DE}$ are equal chords of a circle with center $ O$. Arc $ AB$ is a quarter-circle. Then the ratio of the area of triangle $ CED$ to the area of triangle $ AOB$ is:
<image1>
(A) $\sqrt{2} : 1$
(B) $\sqrt{3} : 1$
(C) $4 : 1$
(D) $3 : 1$
(E) $2 : 1$


Question:
<image1>

Equilateral $ \triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $ T$ and tangent to sides $ AB$ and $ AC$ at points $ P$ and $ Q$. If side $ BC$ has length $ 12$, then segment $ PQ$ has length

Choices:
- $6$
- $6\sqrt{3}$
- $8$
- $8\sqrt{3}$
- $9$

metric geometry - length

Question:
<image1>

Equilateral $ \triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $ T$ and tangent to sides $ AB$ and $ AC$ at points $ P$ and $ Q$. If side $ BC$ has length $ 12$, then segment $ PQ$ has length
(A) $6$
(B) $6\sqrt{3}$
(C) $8$
(D) $8\sqrt{3}$
(E) $9$


Question:
The sum the dots on opposite faces of a die always equals 7. A die rolls as shown below. At the starting point $(A)$ the top face is 3. Which will be the face at the end point $(B)$?
<image1>

Answer: 6

metric geometry - length

Question:
The sum the dots on opposite faces of a die always equals 7. A die rolls as shown below. At the starting point $(A)$ the top face is 3. Which will be the face at the end point $(B)$?
<image1>

Answer: 6


Question:
The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo's house. Eva walks past Chole's house. Which is Eva's house?
<image1>
<image2>

Choices:
- A
- B
- C
- D
- E

logic

Question:
The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo's house. Eva walks past Chole's house. Which is Eva's house?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles.\n<image1>

Answer: rac{36}{13}

Solution: If R is the radius of a great circle,and r-the radius of a one smaller circle,then we have that: R=7r=> the area of a great circle is πR^2=49πr^2, and the area of all the 13 circles is 13πr^2=> area of region is 36πr^2=>36/13 is the answer.

metric geometry - area

Question: Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles.\n<image1>

Answer: rac{36}{13}


Question: In the diagram, $W X Y Z$ is a square, $M$ is the midpoint of $W Z$ and $M N$ is perpendicular to $W Y$. What is the ratio of the area of the shaded triangle $M N Y$ to the area of the square? <image1>

Choices:
- 1:6
- 1:5
- 7:36
- 3:16
- 7:40

Solution: Introduce point $T$, the mid-point of $W Y$. $M T$ is parallel to $Z Y$ and half the length. The area of triangle $W M T$ is $\frac{1}{4}$ of the area of triangle $W Z Y=\frac{1}{8}$ of the area of the square. Also, the area of triangle $W M N$ is $\frac{1}{2}$ of the area of triangle $W M T=\frac{1}{16}$ of the area of the square. The area of triangle $W M Y$ is $\frac{1}{4}$ of the area of the square so the area of triangle NMY is $\left(\frac{1}{4}-\frac{1}{16}\right)=\frac{3}{16}$ of the area of the square. Hence the ratio of the area of triangle $M N Y$ to the area of the square is $3: 16$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-147.jpg?height=345&width=371&top_left_y=2209&top_left_x=1439)  Alternative solution: Suppose that the square has side length $s$ and hence area $s^{2}$. The triangle $W M Y$ has a base of length $\frac{1}{2} s$ and height $s$, and hence area $\frac{1}{2}\left(\frac{1}{2} s \times s\right)=\frac{1}{4} s^{2}$. Triangle $W N M$ is a right-angled isosceles triangle with hypotenuse of length $\frac{1}{2} s$. Let $t$ be the lengths of the other two sides. So the area of WNM is $\frac{1}{2} t^{2}$. By Pythagoras' Theorem $t^{2}+t^{2}=\left(\frac{1}{2} s\right)^{2}$. Therefore $\frac{1}{2} t^{2}=\frac{1}{16} s^{2}$. So the area of the triangle $M N Y$ is $\frac{1}{4} s^{2}-\frac{1}{16} s^{2}=\frac{3}{16} s^{2}$. Hence the ratio of the area of the triangle $M N Y$ to the area of the square is $3: 16$.

metric geometry - area

Question:
In the diagram, $W X Y Z$ is a square, $M$ is the midpoint of $W Z$ and $M N$ is perpendicular to $W Y$. What is the ratio of the area of the shaded triangle $M N Y$ to the area of the square? <image1>
(A) 1:6
(B) 1:5
(C) 7:36
(D) 3:16
(E) 7:40


Question:
Which cloud contains even numbers only?
<image1>

Choices:
- A
- B
- C
- D
- E

arithmetic

Question:
Which cloud contains even numbers only?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
<image1>

Choices:
- 1:1
- 47:43
- 2:1
- 40:13
- 4:1

solid geometry

Question:
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
<image1>
(A) 1:1
(B) 47:43
(C) 2:1
(D) 40:13
(E) 4:1


Question:
How many white squares need to be coloured in black, so that there are exactly twice as many white squares as there are black squares?
<image1>

Answer: 3

counting

Question:
How many white squares need to be coloured in black, so that there are exactly twice as many white squares as there are black squares?
<image1>

Answer: 3


Question:
<image1>
In the picture above we see a cube in two different positions.
The six sides of the cube look like this:
<image2>
Which side is opposite to <image3>?
<image4>

Answer: C

solid geometry

Question:
<image1>
In the picture above we see a cube in two different positions.
The six sides of the cube look like this:
<image2>
Which side is opposite to <image3>?
<image4>

Answer: C


Question:
The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

<image1>

Answer: 6

solid geometry

Question:
The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

<image1>

Answer: 6


Question:
In acute triangle $ABC$, $\angle A = 68^\circ$.  Let $O$ be the circumcenter of triangle $ABC$.  Find $\angle OBC$, in degrees.

<image1>

Answer: 22

Solution:
Since $O$ is the center of the circle passing through $A$, $B$, and $C$, $\angle BOC = 2 \angle BAC = 2 \cdot 68^\circ = 136^\circ$.

[asy]
unitsize(1.5 cm);

pair A, B, C, O;

A = (1,2);
B = (0,0);
C = (3,0);
O = circumcenter(A,B,C);

draw(A--B--C--cycle);
draw(circumcircle(A,B,C));
draw(B--O--C);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
dot("$O$", O, N);
[/asy]

Since $BO = CO$ (both are equal to the circumradius of triangle $ABC$), triangle $BOC$ is isosceles.  Hence, $\angle OBC = (180^\circ - \angle BOC)/2 = (180^\circ - 136^\circ)/2 = \boxed{22^\circ}$.

metric geometry - angle

Question:
In acute triangle $ABC$, $\angle A = 68^\circ$.  Let $O$ be the circumcenter of triangle $ABC$.  Find $\angle OBC$, in degrees.

<image1>

Answer: 22


Question:
Which shape cannot be seen in every picture?
<image1>

Choices:
- A
- B
- C
- D
- E

combinatorial geometry

Question:
Which shape cannot be seen in every picture?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
How many squares can be drawn by joining the dots with line segments?
<image1>

Answer: 4

combinatorics

Question:
How many squares can be drawn by joining the dots with line segments?
<image1>

Answer: 4


Question:
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
<image1>

Answer: 5

metric geometry - angle

Question:
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
<image1>

Answer: 5


Question:
A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
<image1>

Choices:
- $\frac{1}{3}$
- $\frac{2}{5}$
- $\frac{5}{12}$
- $\frac{4}{9}$
- $\frac{1}{2}$

metric geometry - length

Question:
A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
<image1>
(A) $\frac{1}{3}$
(B) $\frac{2}{5}$
(C) $\frac{5}{12}$
(D) $\frac{4}{9}$
(E) $\frac{1}{2}$


Question: What is the largest number of " $\mathrm{T}$ " shaped pieces, as shown, that can be placed on the $4 \times 5$ grid in the diagram, without any overlap of the pieces? <image1>

Answer: 4

Solution: Colour the cells of the grid alternately black and white, as shown in the first diagram. Each "T" shaped piece fits over three cells of one colour and one of the other colour, as shown in the second diagram. Suppose that for one of the two colours, say white, there are three pieces each covering three cells of that colour. Since there are only ten white cells in  ![](https://cdn.mathpix.com/cropped/2023_12_27_cd50dd081de367811fcfg-04.jpg?height=208&width=272&top_left_y=661&top_left_x=1600) the grid, only one white cell is not covered by the three pieces. Hence, as each piece covers at least one white cell, at most one more piece could be placed on the grid. Therefore at most four pieces could be placed on the grid.  Alternatively, if there are no more than two "T" shaped pieces that each cover three white cells and no more than two that cover three black cells, then again there is a maximum of four "T" shaped pieces that could be placed on the grid.  ![](https://cdn.mathpix.com/cropped/2023_12_27_cd50dd081de367811fcfg-04.jpg?height=217&width=277&top_left_y=908&top_left_x=1598)  \begin{tabular}{|l|l|l|l|l|} \hline 1 & & 4 & 4 & 4 \\ \hline 1 & 1 & & 4 & 3 \\ \hline 1 & 2 & & 3 & 3 \\ \hline 2 & 2 & 2 & & 3 \\ \hline \end{tabular}  The third diagram shows one of the many different possible ways in which four pieces could be placed, showing that it is possible to place four pieces on the grid.

combinatorial geometry

Question: What is the largest number of " $\mathrm{T}$ " shaped pieces, as shown, that can be placed on the $4 \times 5$ grid in the diagram, without any overlap of the pieces? <image1>

Answer: 4


Question: In the triangle $F G H$, we can draw a line parallel to its base $F G$, through point $X$ or $Y$. The areas of the shaded regions are the same. The ratio $H X: X F=4: 1$. What is the ratio $H Y: Y F$ ? <image1>

Choices:
- $1: 1$
- $2: 1$
- $3: 1$
- $3: 2$
- $4: 3$

Solution: In the triangle on the left, the unshaded triangle is similar to triangle $F G H$, and is obtained from it by a scale factor of $\frac{4}{5}$. Hence its area is $\left(\frac{4}{5}\right)^{2}=\frac{16}{25}$ of the area of $F G H$. The shaded area is therefore $\frac{9}{25}$ of the area of $F G H$. Hence $H Y: H F=3: 5$ and so $H Y: Y F=3: 2$.

metric geometry - length

Question:
In the triangle $F G H$, we can draw a line parallel to its base $F G$, through point $X$ or $Y$. The areas of the shaded regions are the same. The ratio $H X: X F=4: 1$. What is the ratio $H Y: Y F$ ? <image1>
(A) $1: 1$
(B) $2: 1$
(C) $3: 1$
(D) $3: 2$
(E) $4: 3$


Question:
The area of the wooden square equals $a$. The area of each wooden circle equals $b$. Three circles are lined up as shown in the picture. If we tie together the three circles with a thread as short as possible, without moving them, what is the area inside the thread?
<image1>

Choices:
- 3b
- 2a + b
- a + 2b
- 3a
- a + b

metric geometry - area

Question:
The area of the wooden square equals $a$. The area of each wooden circle equals $b$. Three circles are lined up as shown in the picture. If we tie together the three circles with a thread as short as possible, without moving them, what is the area inside the thread?
<image1>
(A) 3b
(B) 2a + b
(C) a + 2b
(D) 3a
(E) a + b


Question:
Three villages are connected by paths as shown. From Downend to Uphill, the detour via Middleton is $1 \mathrm{~km}$ longer than the direct path. From Downend to Middleton, the detour via Uphill is $5 \mathrm{~km}$ longer than the direct path. From Uphill to Middleton, the detour via Downend is $7 \mathrm{~km}$ longer than the direct path. What is the length of the shortest of the three direct paths between the villages?
<image1>

Choices:
- $1 \mathrm{~km}$
- $2 \mathrm{~km}$
- $3 \mathrm{~km}$
- $4 \mathrm{~km}$
- $5 \mathrm{~km}$

Solution: Let the lengths of the direct paths from Uphill to Middleton, Middleton to Downend and Downend to Uphill be $x \mathrm{~km}, y \mathrm{~km}$ and $z \mathrm{~km}$ respectively. The information in the question tells us that $x+y=z+1, x+z=y+5$ and $y+z=x+7$. When we add these three equations, we obtain $2 x+2 y+2 z=z+y+x+13$ and hence $x+y+z=13$. Therefore $13=2 z+1$, $13=2 y+5$ and $13=2 x+7$, which have solutions $z=6, y=4$ and $x=3$. Hence the length of the shortest of the direct paths is the one from Uphill to Middleton with length $3 \mathrm{~km}$.

metric geometry - length

Question:
Three villages are connected by paths as shown. From Downend to Uphill, the detour via Middleton is $1 \mathrm{~km}$ longer than the direct path. From Downend to Middleton, the detour via Uphill is $5 \mathrm{~km}$ longer than the direct path. From Uphill to Middleton, the detour via Downend is $7 \mathrm{~km}$ longer than the direct path. What is the length of the shortest of the three direct paths between the villages?
<image1>
(A) $1 \mathrm{~km}$
(B) $2 \mathrm{~km}$
(C) $3 \mathrm{~km}$
(D) $4 \mathrm{~km}$
(E) $5 \mathrm{~km}$


Question:
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD=3$, and $ DC=4$. What is the area of $ \triangle{ABC}$?
<image1>

Choices:
- $4\sqrt{3}$
- $7\sqrt{3}$
- $21$
- $14\sqrt{3}$
- $42$

metric geometry - area

Question:
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD=3$, and $ DC=4$. What is the area of $ \triangle{ABC}$?
<image1>
(A) $4\sqrt{3}$
(B) $7\sqrt{3}$
(C) $21$
(D) $14\sqrt{3}$
(E) $42$


Question: The diagram shows a square $P Q R S$ and two equilateral triangles $R S U$ and PST. PQ has length 1 . What is the length of $T U$ ? <image1>

Choices:
- $\sqrt{2}$
- $\frac{\sqrt{3}}{2}$
- $\sqrt{3}$
- $\sqrt{5}-1$
- $\sqrt{6}-1$

Solution: The angles in equilateral triangles are all $60^{\circ}$ so $\angle P S U=90^{\circ}-60^{\circ}=30^{\circ}$, $\angle T S U=30^{\circ}+60^{\circ}=90^{\circ}$ and $U S=1=S T$. Using Pythagoras' theorem on the right-angled triangle $T S U$ we have $T U^{2}=1^{2}+1^{2}=2$ so $T U=\sqrt{2}$.

metric geometry - length

Question:
The diagram shows a square $P Q R S$ and two equilateral triangles $R S U$ and PST. PQ has length 1 . What is the length of $T U$ ? <image1>
(A) $\sqrt{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\sqrt{3}$
(D) $\sqrt{5}-1$
(E) $\sqrt{6}-1$


Question:
The degree measure of angle $A$  is
<image1>

Answer: 30

metric geometry - angle

Question:
The degree measure of angle $A$  is
<image1>

Answer: 30


Question:
Inside the gray square there are three white squares and the numbers inside them indicate their areas. The white squares have sides parallel to the sides of the gray square. If the area of the gray square is 81, what is the area of the gray area not covered by the white squares?
<image1>

Answer: 52

metric geometry - area

Question:
Inside the gray square there are three white squares and the numbers inside them indicate their areas. The white squares have sides parallel to the sides of the gray square. If the area of the gray square is 81, what is the area of the gray area not covered by the white squares?
<image1>

Answer: 52


Question:
You write a number in each square as shown in the square figure. Then, the number $x$ cannot be:
<image1>

Answer: 128

algebra

Question:
You write a number in each square as shown in the square figure. Then, the number $x$ cannot be:
<image1>

Answer: 128


Question:
In the box are seven blockss. You want to rearrange the blocks so that another block can placed. What is the minimum number of blocks that have to be moved?
<image1>

Answer: 3

combinatorial geometry

Question:
In the box are seven blockss. You want to rearrange the blocks so that another block can placed. What is the minimum number of blocks that have to be moved?
<image1>

Answer: 3


Question:
Georg goes to the circus with his father. They have the seat numbers 71 and 72 . Which arrow do they have to follow to get to their seats?
<image1>
<image2>

Choices:
- (A)
- (B)
- (C)
- (D)
- (E)

arithmetic

Question:
Georg goes to the circus with his father. They have the seat numbers 71 and 72 . Which arrow do they have to follow to get to their seats?
<image1>
<image2>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

<image1>

Choices:
- $\frac{2}{\sqrt{\pi}}$
- $\frac{1+\sqrt{2}}{2}$
- $\frac{3}{2}$
- $\sqrt{3}$
- $\sqrt{\pi}$

metric geometry - length

Question:
A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

<image1>
(A) $\frac{2}{\sqrt{\pi}}$
(B) $\frac{1+\sqrt{2}}{2}$
(C) $\frac{3}{2}$
(D) $\sqrt{3}$
(E) $\sqrt{\pi}$


Question: In isosceles triangle $ABC$, angle $BAC$ and angle $BCA$ measure 35 degrees. What is the measure of angle $CDA$? <image1>

Answer: 70

Solution: Angles $BAC$ and $BCA$ are each inscribed angles, so each one is equal to half of the measure of the arc they subtend. Therefore, the measures of arcs $AB$ and $BC$ are each 70 degrees, and together, the measure of arc $ABC$ is 140 degrees. Notice that angle $CDA$ is also an inscribed angle, and it subtends arc $ABC$, so $m\angle CDA = \frac{1}{2} (\text{arc } ABC) = (1/2)(140) = \boxed{70}$ degrees.

metric geometry - angle

Question: In isosceles triangle $ABC$, angle $BAC$ and angle $BCA$ measure 35 degrees. What is the measure of angle $CDA$? <image1>

Answer: 70


Question:
Riki wants to write one number in each of the seven sections of the diagram pictured. Two zones are adjacent if they share a part of their outline. The number in each zone should be the sum of all numbers of its adjacent zones. Riki has already placed numbers in two zones. Which number does she need to write in the zone marked "?".
<image1>

Answer: 6

algebra

Question:
Riki wants to write one number in each of the seven sections of the diagram pictured. Two zones are adjacent if they share a part of their outline. The number in each zone should be the sum of all numbers of its adjacent zones. Riki has already placed numbers in two zones. Which number does she need to write in the zone marked "?".
<image1>

Answer: 6


Question:
A square with perimeter $48 \mathrm{~cm}$ is cut into two equally big pieces with one cut. They are fitted together to make a rectangle as shown in the diagram. How big is the perimeter of that rectangle?
<image1>

Choices:
- $24 \mathrm{~cm}$
- $30 \mathrm{~cm}$
- $48 \mathrm{~cm}$
- $60 \mathrm{~cm}$
- $72 \mathrm{~cm}$

transformation geometry

Question:
A square with perimeter $48 \mathrm{~cm}$ is cut into two equally big pieces with one cut. They are fitted together to make a rectangle as shown in the diagram. How big is the perimeter of that rectangle?
<image1>
(A) $24 \mathrm{~cm}$
(B) $30 \mathrm{~cm}$
(C) $48 \mathrm{~cm}$
(D) $60 \mathrm{~cm}$
(E) $72 \mathrm{~cm}$


Question:
The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E?

<image1>

Answer: 20

statistics

Question:
The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E?

<image1>

Answer: 20


Question:
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
<image1>

Answer: 244

combinatorics

Question:
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
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Answer: 244