Question:
Peter has bought a rug that is $36 \mathrm{dm}$ wide and $60 \mathrm{dm}$ long. On the rug you can see squares that contain either a sun or a moon, as shown in the picture. As you can see there are exactly nine squares along the width of the rug. The total length of the rug cannot be seen. How many moons would you see, if you could see the entire rug?
<image1>

Answer: 67

arithmetic

Question:
Peter has bought a rug that is $36 \mathrm{dm}$ wide and $60 \mathrm{dm}$ long. On the rug you can see squares that contain either a sun or a moon, as shown in the picture. As you can see there are exactly nine squares along the width of the rug. The total length of the rug cannot be seen. How many moons would you see, if you could see the entire rug?
<image1>

Answer: 67


Question:
The solid shown has a square base of side length $s$.  The upper edge is parallel to the base and has length $2s$.  All other edges have length $s$.  Given that $s = 6 \sqrt{2}$, what is the volume of the solid?
<image1>

Answer: 288

solid geometry

Question:
The solid shown has a square base of side length $s$.  The upper edge is parallel to the base and has length $2s$.  All other edges have length $s$.  Given that $s = 6 \sqrt{2}$, what is the volume of the solid?
<image1>

Answer: 288


Question:
Daniel wants to make a complete square using pieces only like those shown. What is the minimum number of pieces he must use?
<image1>

Answer: 20

combinatorial geometry

Question:
Daniel wants to make a complete square using pieces only like those shown. What is the minimum number of pieces he must use?
<image1>

Answer: 20


Question:
Paul wants to write a positive whole number onto every tile in the number wall shown, so that every number is equal to the sum of the two numbers on the tiles that are directly below. What is the maximum number of odd numbers he can write on the tiles?
<image1>

Answer: 14

combinatorics

Question:
Paul wants to write a positive whole number onto every tile in the number wall shown, so that every number is equal to the sum of the two numbers on the tiles that are directly below. What is the maximum number of odd numbers he can write on the tiles?
<image1>

Answer: 14


Question:
An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$)
<image1>

Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals:

Choices:
- 180
- 360
- 180(n+2)
- 180(n-2)
- 180(n-4)

metric geometry - angle

Question:
An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$)
<image1>

Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals:
(A) 180
(B) 360
(C) 180(n+2)
(D) 180(n-2)
(E) 180(n-4)


Question:
The kangaroos $A, B, C, D$ and $E$ sit in this order in a clockwise direction around a round table. After a bell sounds all but one kangaroo change seats with a neighbour. Afterwards they sit in the following order in a clockwise direction: A, E, B, D, C. Which kangaroo did not change places?
<image1>

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

graph theory

Question:
The kangaroos $A, B, C, D$ and $E$ sit in this order in a clockwise direction around a round table. After a bell sounds all but one kangaroo change seats with a neighbour. Afterwards they sit in the following order in a clockwise direction: A, E, B, D, C. Which kangaroo did not change places?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches.
<image1>
Then $PB$, in inches, equals:

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

metric geometry - length

Question:
$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches.
<image1>
Then $PB$, in inches, equals:
(A) $2\sqrt{3}$
(B) $3\sqrt{2}$
(C) $3\sqrt{3}$
(D) $4\sqrt{2}$
(E) $2$


Question: On the table there is a tower made of blocks numbered from 1 to 90 , as shown on the left of the diagram. Yett takes blocks from the top of the tower, three at a time, to build a new tower, as shown on the right of the diagram. How many blocks will be between blocks 39 and 40 when he has finished building the new tower? <image1>

Answer: 4

Solution: If $n$ is divisible by 3 , then block $n$ will be be moved with the two blocks underneath it, $n-1$ and $n-2$.  Since 42 is divisible by 3, block 42 will be moved with blocks number 41 and 40 underneath block 42. So now our new tower from top down looks like: 42, 41, 40, 45, 44, . . . .  Yett would then next move the three blocks that are on the top of what remains of the original tower, namely 39,38 and 37 . Adding these to our previous set gives the order as: $\mathbf{3 9}, 38,37,42,41, \mathbf{4 0}, 45,44, \ldots$.  It follows that between blocks with numbers 39 and 40 there will be 4 blocks.

algebra

Question: On the table there is a tower made of blocks numbered from 1 to 90 , as shown on the left of the diagram. Yett takes blocks from the top of the tower, three at a time, to build a new tower, as shown on the right of the diagram. How many blocks will be between blocks 39 and 40 when he has finished building the new tower? <image1>

Answer: 4


Question:
Paulo took a rectangular sheet of paper, yellow on one side and green on the other side and, with several folds shown in the dotted lines in the figure below, made a little paper plane. To give the airplane a charm, Paulo made a circular hole, marked on the last figure.
<image1>
After playing a lot with the plane, Paulo unfolded the sheet and realized that there were several holes in it. How many holes did he count?

Answer: 8

descriptive geometry

Question:
Paulo took a rectangular sheet of paper, yellow on one side and green on the other side and, with several folds shown in the dotted lines in the figure below, made a little paper plane. To give the airplane a charm, Paulo made a circular hole, marked on the last figure.
<image1>
After playing a lot with the plane, Paulo unfolded the sheet and realized that there were several holes in it. How many holes did he count?

Answer: 8


Question:
In the picture there are four overlapping squares with sides 11, 9, 7 and 5 long. How much greater is the sum of the two grey areas than the sum of the two black areas?
<image1>

Answer: 64

metric geometry - area

Question:
In the picture there are four overlapping squares with sides 11, 9, 7 and 5 long. How much greater is the sum of the two grey areas than the sum of the two black areas?
<image1>

Answer: 64


Question:
Elliott Farms has a silo for storage. The silo is a right circular cylinder topped by a right circular cone, both having the same radius. The height of the cone is half the height of the cylinder. The diameter of the base of the silo is 10 meters and the height of the entire silo is 27 meters. What is the volume, in cubic meters, of the silo? Express your answer in terms of $\pi$.

<image1>

Answer: 525\pi

Solution: To begin, see that if the ratio of the height of the cone to the height of the cylinder is 1:2, then the ratio of the cone height to the entire silo height is 1:3.  Therefore, the height of the cone is $27/3=9$ meters and the height of the cylinder is $18$ meters.  We can now use the formulas for the volume of a cylinder and the volume of a cone, with our given radius of 5: $$V_{cone}=\frac{1}{3}\cdot b \cdot h=\frac{1}{3}\cdot (\pi\cdot 5^2)\cdot 9=75\pi$$$$V_{cylinder}=\pi r^2\cdot h=\pi 5^2\cdot 18=450\pi$$$$V_{silo}=V_{cone}+V_{cylinder}=75\pi+450\pi=\boxed{525\pi}.$$

solid geometry

Question:
Elliott Farms has a silo for storage. The silo is a right circular cylinder topped by a right circular cone, both having the same radius. The height of the cone is half the height of the cylinder. The diameter of the base of the silo is 10 meters and the height of the entire silo is 27 meters. What is the volume, in cubic meters, of the silo? Express your answer in terms of $\pi$.

<image1>

Answer: 525\pi


Question:
Two squares $9 \mathrm{~cm} \times 9 \mathrm{~cm}$ overlap so as to form a rectangle $9 \mathrm{~cm} \times 13 \mathrm{~cm}$ as shown. Find the area of the region in which the two squares overlap.
<image1>

Choices:
- $36 \mathrm{~cm}^{2}$
- $45 \mathrm{~cm}^{2}$
- $54 \mathrm{~cm}^{2}$
- $63 \mathrm{~cm}^{2}$
- $72 \mathrm{~cm}^{2}$

metric geometry - area

Question:
Two squares $9 \mathrm{~cm} \times 9 \mathrm{~cm}$ overlap so as to form a rectangle $9 \mathrm{~cm} \times 13 \mathrm{~cm}$ as shown. Find the area of the region in which the two squares overlap.
<image1>
(A) $36 \mathrm{~cm}^{2}$
(B) $45 \mathrm{~cm}^{2}$
(C) $54 \mathrm{~cm}^{2}$
(D) $63 \mathrm{~cm}^{2}$
(E) $72 \mathrm{~cm}^{2}$


Question:
Jim and Ben are sitting in a ferris wheel (see picture on the right). The ferris wheel is turning. Now Ben is in the position where Jim was beforehand. Where is Jim now?
<image1>
<image2>

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

transformation geometry

Question:
Jim and Ben are sitting in a ferris wheel (see picture on the right). The ferris wheel is turning. Now Ben is in the position where Jim was beforehand. Where is Jim now?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Belinda is making patterns with toothpicks according to the schema of the figure. How many toothpicks does Belinda add to the 30th pattern to make the 31 st?
<image1>

Answer: 124

algebra

Question:
Belinda is making patterns with toothpicks according to the schema of the figure. How many toothpicks does Belinda add to the 30th pattern to make the 31 st?
<image1>

Answer: 124


Question:
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$?
<image1>

Choices:
- $\frac{1}{12}$
- $\frac{1}{9}$
- $\frac{1}{8}$
- $\frac{1}{6}$
- $\frac{1}{4}$

solid geometry

Question:
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$?
<image1>
(A) $\frac{1}{12}$
(B) $\frac{1}{9}$
(C) $\frac{1}{8}$
(D) $\frac{1}{6}$
(E) $\frac{1}{4}$


Question:
Betty keeps drawing three different figures in the same order. Which figure should be the next?
<image1>
<image2>

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

transformation geometry

Question:
Betty keeps drawing three different figures in the same order. Which figure should be the next?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
By shooting two arrows at the shown target on the wall, how many different scores can we obtain?
<image1>

Answer: 9

combinatorics

Question:
By shooting two arrows at the shown target on the wall, how many different scores can we obtain?
<image1>

Answer: 9


Question:
In triangle $ABC$, $\angle BAC = 72^\circ$.  The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively.  Find $\angle EDF$, in degrees.

<image1>

Answer: 54

Solution:
Since $BD$ and $BF$ are tangents from the same point to the same circle, $BD = BF$.  Hence, triangle $BDF$ is isosceles, and $\angle BDF = (180^\circ - \angle B)/2$.  Similarly, triangle $CDE$ is isosceles, and $\angle CDE = (180^\circ - \angle C)/2$.

Hence, \begin{align*}
\angle FDE &= 180^\circ - \angle BDF - \angle CDE \\
&= 180^\circ - \frac{180^\circ - \angle B}{2} - \frac{180^\circ - \angle C}{2} \\
&= \frac{\angle B + \angle C}{2}.
\end{align*} But $\angle A + \angle B + \angle C = 180^\circ$, so \[\frac{\angle B + \angle C}{2} = \frac{180^\circ - \angle A}{2} = \frac{180^\circ - 72^\circ}{2} = \boxed{54^\circ}.\]

metric geometry - angle

Question:
In triangle $ABC$, $\angle BAC = 72^\circ$.  The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively.  Find $\angle EDF$, in degrees.

<image1>

Answer: 54


Question:
<image1>

Five cards are lying on a table as shown.  Each card has a letter on one side and a whole number on the other side.  Jane said, "If a vowel is on one side of any card, then an even number is on the other side."  Mary showed Jane was wrong by turning over one card.  Which card did Mary turn over?

Choices:
- $3$
- $4$
- $6$
- $\text{P}$
- $\text{Q}$

logic

Question:
<image1>

Five cards are lying on a table as shown.  Each card has a letter on one side and a whole number on the other side.  Jane said, "If a vowel is on one side of any card, then an even number is on the other side."  Mary showed Jane was wrong by turning over one card.  Which card did Mary turn over?
(A) $3$
(B) $4$
(C) $6$
(D) $\text{P}$
(E) $\text{Q}$


Question:
One of the five coins $A, B, C, D$ or $E$ shall be placed in an empty square so that there are exactly two coins in each row and in each column. Which coin should be moved?
<image1>

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

logic

Question:
One of the five coins $A, B, C, D$ or $E$ shall be placed in an empty square so that there are exactly two coins in each row and in each column. Which coin should be moved?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
<image1>

Choices:
- 1:3
- 3:8
- 5:12
- 7:16
- 4:9

metric geometry - area

Question:
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
<image1>
(A) 1:3
(B) 3:8
(C) 5:12
(D) 7:16
(E) 4:9


Question:
In right triangle $XYZ$, shown below, what is $\sin{X}$?

<image1>

Answer: \frac{3}{5}

Solution: From the Pythagorean Theorem, we have \begin{align*}XY^2+YZ^2&=XZ^2 \\ \Rightarrow\qquad{YZ}&=\sqrt{XZ^2-XY^2} \\ &=\sqrt{10^2-8^2} \\ &=\sqrt{36} \\ &=6.\end{align*}Therefore, $\sin{X}=\frac{YZ}{XZ}={\frac{6}{10}}=\boxed{\frac{3}{5}}$.

metric geometry - angle

Question:
In right triangle $XYZ$, shown below, what is $\sin{X}$?

<image1>

Answer: \frac{3}{5}


Question:
In a box-shaped water tank with dimensions $4 \mathrm{~m} \times 2 \mathrm{~m} \times 1 \mathrm{~m}$, the height of the water is $25 \mathrm{~cm}$. The tank is then turned on its side (see picture on the right). How high is the water in the tank now?
<image1>

Choices:
- $25 \mathrm{~cm}$
- $50 \mathrm{~cm}$
- $75 \mathrm{~cm}$
- $1 \mathrm{~m}$
- $1.25 \mathrm{~m}$

solid geometry

Question:
In a box-shaped water tank with dimensions $4 \mathrm{~m} \times 2 \mathrm{~m} \times 1 \mathrm{~m}$, the height of the water is $25 \mathrm{~cm}$. The tank is then turned on its side (see picture on the right). How high is the water in the tank now?
<image1>
(A) $25 \mathrm{~cm}$
(B) $50 \mathrm{~cm}$
(C) $75 \mathrm{~cm}$
(D) $1 \mathrm{~m}$
(E) $1.25 \mathrm{~m}$


Question:
In the figure on the right a few of the small squares will be painted grey. In so doing no square <image1> that is made from four small grey squares must appear. At most how many of the squares in the figure can be painted grey?
<image2>

Answer: 21

graph theory

Question:
In the figure on the right a few of the small squares will be painted grey. In so doing no square <image1> that is made from four small grey squares must appear. At most how many of the squares in the figure can be painted grey?
<image2>

Answer: 21


Question:
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct?

<image1>

Choices:
- B = W
- W = R
- B = R
- 3B = 2R
- 2R = W

metric geometry - area

Question:
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct?

<image1>
(A) B = W
(B) W = R
(C) B = R
(D) 3B = 2R
(E) 2R = W


Question:
Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC=  DE = FG = HA = 11$ is formed by removing four $6-8-10$ triangles from the corners of a $23\times 27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{HA}$, and partition the octagon into $7$ triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these $7$ triangles.

<image1>

Answer: 184

metric geometry - area

Question:
Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC=  DE = FG = HA = 11$ is formed by removing four $6-8-10$ triangles from the corners of a $23\times 27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{HA}$, and partition the octagon into $7$ triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these $7$ triangles.

<image1>

Answer: 184


Question:
Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?

<image1>

Choices:
- <image2>
- <image3>
- <image4>
- <image5>
- <image6>

statistics

Question:
Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?

<image1>
(A) <image2>
(B) <image3>
(C) <image4>
(D) <image5>
(E) <image6>


Question:
Mike and Jake play darts. Each of them throws three darts. Who won, and by how many points?
Mike: <image1>
Jake: <image2>

Choices:
- Mike won. He had 3 points more.
- Jake won. He had 4 points more.
- Mike won. He had 2 points more.
- Jake won. He had 2 points more.
- Mike won. He had 4 points more.

arithmetic

Question:
Mike and Jake play darts. Each of them throws three darts. Who won, and by how many points?
Mike: <image1>
Jake: <image2>
(A) Mike won. He had 3 points more.
(B) Jake won. He had 4 points more.
(C) Mike won. He had 2 points more.
(D) Jake won. He had 2 points more.
(E) Mike won. He had 4 points more.


Question: In the diagram, the four points have coordinates $A(0,1)$, $B(1,3)$, $C(5,2)$, and $D(4,0)$.  What is the area of quadrilateral $ABCD$? <image1>

Answer: 9

Solution:
We draw a horizontal line through $B$ (meeting the $y$-axis at $P$) and a vertical line through $C$ (meeting the $x$-axis at $Q$).  Suppose the point of intersection of these two lines is $R$. [asy]
// draw axis
real low = -1; real high = 6;
draw((low, 0)--(high, 0), Arrow); draw((0, low)--(0, high - 1.5), Arrow);
label("$x$", (high, 0), E); label("$y$", (0, high - 1.5), N);

// draw quadrilateral
pair a = (0, 1); pair b = (1, 3); pair c = (5, 2); pair d = (4, 0);
draw(a--b--c--d--cycle);
label("$A$", a, W); label("$B$", b, N); label("$C$", c, E); label("$D$", d, S);

// add extra points/lines
pair p = (0, b.y); pair q = (c.x, 0); pair r = p + q;
label("$P$", p, W); label("$Q$", q, S); label("$R$", r, NE);
draw(p--r--q);
[/asy] We know that $P$ has coordinates $(0,3)$ (since $B$ has $y$-coordinate 3) and $Q$ has coordinates $(5,0)$ (since $C$ has $x$-coordinate 5), so $R$ has coordinates $(5,3)$.

Using the given coordinates, $OA=1$, $AP=2$, $PB=1$, $BR=4$, $RC=1$, $CQ=2$, $QD=1$, and $DO=4$.

The area of $ABCD$ equals the area of $PRQO$ minus the areas of triangles $APB$, $BRC$, $CQD$, and $DOA$.

$PRQO$ is a rectangle, so it has area $3 \times 5 = 15$.

Triangles $APB$ and $CQD$ have bases $PB$ and $QD$ of length 1 and heights $AP$ and $CQ$ of length 2, so each has area $$\frac{1}{2}(1)(2)=1.$$Triangles $BRC$ and $DOA$ have bases $BR$ and $DO$ of length 4 and heights $CR$ and $AO$ of length 1, so each has area $$\frac{1}{2}(4)(1)=2.$$Thus, the area of $ABCD$ is $$15 -1-1-2-2=\boxed{9}.$$(Alternatively, we could notice that $ABCD$ is a parallelogram.  Therefore, if we draw the diagonal $AC$, the area is split into two equal pieces.  Dropping a perpendicular from $C$ to $Q$ on the $x$-axis produces a trapezoid $ACQO$ from which only two triangles need to be removed to determine half of the area of $ABCD$.)

analytic geometry

Question: In the diagram, the four points have coordinates $A(0,1)$, $B(1,3)$, $C(5,2)$, and $D(4,0)$.  What is the area of quadrilateral $ABCD$? <image1>

Answer: 9


Question:
With how many ways one can get a number 2006 while following the arrows on the figure?
<image1>

Answer: 8

graph theory

Question:
With how many ways one can get a number 2006 while following the arrows on the figure?
<image1>

Answer: 8


Question:
A garden of a square shape is divided into a pool (P), a flowerbed (F), a lawn (L) and a sandpit (S) (see the picture). The lawn and the flowerbed are of a square shape. The perimeter of the lawn is $20 \mathrm{~m}$, the perimeter of the flowerbed is $12 \mathrm{~m}$. What is the perimeter of the pool?
<image1>

Choices:
- $10 \mathrm{~m}$
- $12 \mathrm{~m}$
- $14 \mathrm{~m}$
- $16 \mathrm{~m}$
- $18 \mathrm{~m}$

metric geometry - length

Question:
A garden of a square shape is divided into a pool (P), a flowerbed (F), a lawn (L) and a sandpit (S) (see the picture). The lawn and the flowerbed are of a square shape. The perimeter of the lawn is $20 \mathrm{~m}$, the perimeter of the flowerbed is $12 \mathrm{~m}$. What is the perimeter of the pool?
<image1>
(A) $10 \mathrm{~m}$
(B) $12 \mathrm{~m}$
(C) $14 \mathrm{~m}$
(D) $16 \mathrm{~m}$
(E) $18 \mathrm{~m}$


Question:
Which of the following graphs represents the solution set of $(x-|x|)^{2}+(y-|y|)^{2}=4$?
<image1>

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

analytic geometry

Question:
Which of the following graphs represents the solution set of $(x-|x|)^{2}+(y-|y|)^{2}=4$?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Julia folds the paper net pictured on the right, into a cube. Which number is on the face that is opposite to the face with the number 3?
<image1>

Answer: 6

descriptive geometry

Question:
Julia folds the paper net pictured on the right, into a cube. Which number is on the face that is opposite to the face with the number 3?
<image1>

Answer: 6


Question: The digits 0 to 9 can be formed using matchsticks (see diagram). How many different positive whole numbers can be formed this way with exactly 6 matchsticks? <image1>

Answer: 6

combinatorics

Question: The digits 0 to 9 can be formed using matchsticks (see diagram). How many different positive whole numbers can be formed this way with exactly 6 matchsticks? <image1>

Answer: 6


Question: The diagram shows three squares of side-length $3 \mathrm{~cm}, 5 \mathrm{~cm}$ and $8 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded trapezium? <image1>

Choices:
- $13$
- $\frac{55}{4}$
- $\frac{61}{4}$
- $\frac{65}{4}$
- $\frac{69}{4}$

Solution: Let the lengths of the vertical sides of the shaded trapezium be $p$ and $q$. Using similar triangles, $\frac{p}{3}=\frac{q}{3+5}=\frac{8}{3+5+8}$. Hence $p=\frac{3}{2}$ and $q=4$.  Therefore the area of the trapezium is $\frac{1}{2} \times\left(\frac{3}{2}+4\right) \times 5$ that is $\frac{55}{4} \mathrm{~cm}^{2}$.

metric geometry - area

Question:
The diagram shows three squares of side-length $3 \mathrm{~cm}, 5 \mathrm{~cm}$ and $8 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded trapezium? <image1>
(A) $13$
(B) $\frac{55}{4}$
(C) $\frac{61}{4}$
(D) $\frac{65}{4}$
(E) $\frac{69}{4}$


Question:
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?

<image1>

Answer: 6

statistics

Question:
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?

<image1>

Answer: 6


Question:
Both the figures on the right were made out of the same 5 pieces. The rectangle has dimensions $5 \mathrm{~cm} \times 10 \mathrm{~cm}$. The other pieces are quarter circles with 2 different sized radii. What is the difference between the perimeters of the two figures?
<image1>

Choices:
- $2.5 \mathrm{~cm}$
- $5 \mathrm{~cm}$
- $10 \mathrm{~cm}$
- $15 \mathrm{~cm}$
- $20 \mathrm{~cm}$

combinatorial geometry

Question:
Both the figures on the right were made out of the same 5 pieces. The rectangle has dimensions $5 \mathrm{~cm} \times 10 \mathrm{~cm}$. The other pieces are quarter circles with 2 different sized radii. What is the difference between the perimeters of the two figures?
<image1>
(A) $2.5 \mathrm{~cm}$
(B) $5 \mathrm{~cm}$
(C) $10 \mathrm{~cm}$
(D) $15 \mathrm{~cm}$
(E) $20 \mathrm{~cm}$


Question:
In each of the squares in the grid, one of the letters $P, Q, R$ and $S$ must be entered in such a way that adjacent squares (whether connected by an edge or just a corner) do not contain the same letter. Some of the letters have already been entered as shown.
What are the possibilities for the letter in the shaded square?
<image1>

Choices:
- only $Q$
- only $R$
- only $S$
- either $R$ or $S$, but no others
- it is impossible to complete the grid

Solution: It is clear that there is a unique way to complete the top three rows, as shown on the right (start in the second square of the third row). Thereafter it is possible to complete the fourth row with $R$ and $S$ alternating and the fifth row $Q P Q P Q$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-098.jpg?height=274&width=297&top_left_y=1165&top_left_x=1508)

logic

Question:
In each of the squares in the grid, one of the letters $P, Q, R$ and $S$ must be entered in such a way that adjacent squares (whether connected by an edge or just a corner) do not contain the same letter. Some of the letters have already been entered as shown.
What are the possibilities for the letter in the shaded square?
<image1>
(A) only $Q$
(B) only $R$
(C) only $S$
(D) either $R$ or $S$, but no others
(E) it is impossible to complete the grid


Question:
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?

<image1>

Answer: 5

statistics

Question:
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?

<image1>

Answer: 5


Question:
Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.

<image1>

Choices:
- $03$
- $1006$
- $504$
- $1507$
- $1510$

algebra

Question:
Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.

<image1>
(A) $03$
(B) $1006$
(C) $504$
(D) $1507$
(E) $1510$


Question:
Five equally big square pieces of card are placed on a table on top of each other. The picture on the side is created this way. The cards are collected up from top to bottom. In which order are they collected?
<image1>

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

descriptive geometry

Question:
Five equally big square pieces of card are placed on a table on top of each other. The picture on the side is created this way. The cards are collected up from top to bottom. In which order are they collected?
<image1>
(A) 5-4-3-2-1
(B) 5-2-3-4-1
(C) 5-4-2-3-1
(D) 5-3-2-1-4
(E) 5-2-3-1-4


Question:
Kamilla wrote down all of the numbers from 1-100 one after the other in a table with 5 columns. A part of the table is shown. Her brother cut out a piece of the table and erased some of the numbers. Which of the following could this piece have been?
<image1>
<image2>

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

algebra

Question:
Kamilla wrote down all of the numbers from 1-100 one after the other in a table with 5 columns. A part of the table is shown. Her brother cut out a piece of the table and erased some of the numbers. Which of the following could this piece have been?
<image1>
<image2>
(A) A)
(B) B)
(C) C)
(D) D)
(E) E)


Question: Suppose Annie the Ant is walking on a regular icosahedron (as shown). She starts on point $A$ and will randomly create a path to go to point $Z$ which is the point directly opposite to $A$. Every move she makes never moves further from Z, and she has equal probability to go down every valid move. What is the expected number of moves she can make?\n<image1>

Answer: 6

Solution: Lets categorize the vertices into 4 levels, defined by distance to $A$. Due to the condition that she never moves further from Z, she can never go to a higher level, so she must stay on her level or progress to a lower level. At the first level, she can only make a move to level two, which is 1 move. At level two, she always has 4 moves she can make, 2 of them move down. At level three, she always has 3 moves she can make, 1 of which moves down.\nLet $E_n$ be the expected number of moves to move from level $n$ to $n+1$. We have $E_n=1$,\n$$\nE_2=\frac{1}{2} \cdot 1+\frac{1}{2} \cdot\left(E_2+1\right) \text { and } E_3=\frac{1}{3} \cdot 1+\frac{2}{3} \cdot\left(E_3\n1\right)\n$$\n\nThis solves to $E_2=2, E_3=3$. The final expected value is, by linearity of expected value, 6 .

solid geometry

Question: Suppose Annie the Ant is walking on a regular icosahedron (as shown). She starts on point $A$ and will randomly create a path to go to point $Z$ which is the point directly opposite to $A$. Every move she makes never moves further from Z, and she has equal probability to go down every valid move. What is the expected number of moves she can make?\n<image1>

Answer: 6


Question:
Which of the 5 pictures shows a part of this chain?
<image1>
<image2>

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

combinatorial geometry

Question:
Which of the 5 pictures shows a part of this chain?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$.  If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals

<image1>

Choices:
- $2$
- $\frac{5}{2}$
- $\frac{5}{2} - \sin \theta$
- $\frac{5}{2} - \frac{1}{2} \sin \theta$
- $\frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$

metric geometry - length

Question:
In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$.  If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals

<image1>
(A) $2$
(B) $\frac{5}{2}$
(C) $\frac{5}{2} - \sin \theta$
(D) $\frac{5}{2} - \frac{1}{2} \sin \theta$
(E) $\frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$


Question:
We see in the diagram at the right a piece of the graphic of the function
$$
f(x)=a x^{3}+b x^{2}+c x+d.
$$
What is the value of $b$?
<image1>

Answer: -2

analytic geometry

Question:
We see in the diagram at the right a piece of the graphic of the function
$$
f(x)=a x^{3}+b x^{2}+c x+d.
$$
What is the value of $b$?
<image1>

Answer: -2


Question:
In the picture $A B C D$ is a rectangle with $A B=16, B C=12$. Let $E$ be such a point that $A C \perp C E, C E=15$. If $F$ is the point of intersection of segments $A E$ and $C D$, then the area of the triangle $A C F$ is equal to
<image1>

Answer: 75

metric geometry - area

Question:
In the picture $A B C D$ is a rectangle with $A B=16, B C=12$. Let $E$ be such a point that $A C \perp C E, C E=15$. If $F$ is the point of intersection of segments $A E$ and $C D$, then the area of the triangle $A C F$ is equal to
<image1>

Answer: 75


Question:
In the picture any letter stands for some digit (different letters for different digits, equal letters for equal digits). Which digit is $\mathrm{K}$?
<image1>

Answer: 9

algebra

Question:
In the picture any letter stands for some digit (different letters for different digits, equal letters for equal digits). Which digit is $\mathrm{K}$?
<image1>

Answer: 9


Question:
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

<image1>

Answer: 9

statistics

Question:
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

<image1>

Answer: 9


Question: The diagram shows a shaded quadrilateral $P Q R S$ drawn on a grid. Each cell of the grid has sides of length $2 \mathrm{~cm}$. What is the area of quadrilateral $P Q R S$ ? <image1>

Choices:
- $96 \mathrm{~cm}^{2}$
- $84 \mathrm{~cm}^{2}$
- $76 \mathrm{~cm}^{2}$
- $88 \mathrm{~cm}^{2}$
- $104 \mathrm{~cm}^{2}$

Solution: Surround the quadrilateral $P Q R S$ by a rectangle with sides parallel to the grid lines as shown. The area of the rectangle is $14 \times 10=140 \mathrm{~cm}^{2}$. The area of quadrilateral $P Q R S$ can be calculated by subtracting from this the sum of the areas of the four triangles and one square that lie outside $P Q R S$ but inside the rectangle from the area of the rectangle. This gives the area of $P Q R S$ as $140-\frac{1}{2} \times 14 \times 2-\frac{1}{2} \times 8 \times 6-\frac{1}{2} \times 6 \times 2-2 \times 2-\frac{1}{2} \times 8 \times 2=140-14-24-6-4-8$ $=84 \mathrm{~cm}^{2}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-164.jpg?height=263&width=331&top_left_y=354&top_left_x=1519)

combinatorial geometry

Question:
The diagram shows a shaded quadrilateral $P Q R S$ drawn on a grid. Each cell of the grid has sides of length $2 \mathrm{~cm}$. What is the area of quadrilateral $P Q R S$ ? <image1>
(A) $96 \mathrm{~cm}^{2}$
(B) $84 \mathrm{~cm}^{2}$
(C) $76 \mathrm{~cm}^{2}$
(D) $88 \mathrm{~cm}^{2}$
(E) $104 \mathrm{~cm}^{2}$


Question:
There are eight kangaroos in a row, as seen in the picture.
<image1>
Two kangaroos, that are standing next to each other and that are looking into each others eyes, are changing places by hopping past each other. This is carried out until no more jumps are possible. How often did a change of places occur?

Answer: 13

counting

Question:
There are eight kangaroos in a row, as seen in the picture.
<image1>
Two kangaroos, that are standing next to each other and that are looking into each others eyes, are changing places by hopping past each other. This is carried out until no more jumps are possible. How often did a change of places occur?

Answer: 13


Question:
The little caterpillar <image1> rolls up to go to sleep. What could it look like then?
<image2>

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

transformation geometry

Question:
The little caterpillar <image1> rolls up to go to sleep. What could it look like then?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
<image1>

In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals

Choices:
- $30^\circ$
- $40^\circ$
- $50^\circ$
- $65^\circ$
- $\text{none of these}$

metric geometry - angle

Question:
<image1>

In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
(A) $30^\circ$
(B) $40^\circ$
(C) $50^\circ$
(D) $65^\circ$
(E) $\text{none of these}$


Question:
Misty the cat has five kittens: two of them are striped, one spotty, the rest of them are absolutely white. In which picture can we see the kittens of Misty, knowing that the ears of one of them are of different colour?
<image1>

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

logic

Question:
Misty the cat has five kittens: two of them are striped, one spotty, the rest of them are absolutely white. In which picture can we see the kittens of Misty, knowing that the ears of one of them are of different colour?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The diagram shows an object made up of 12 dice glued-together. The object is dipped into some colour so that the entire outside is coloured in this new colour. How many of the small dice will have exactly four faces coloured in?
<image1>

Answer: 10

solid geometry

Question:
The diagram shows an object made up of 12 dice glued-together. The object is dipped into some colour so that the entire outside is coloured in this new colour. How many of the small dice will have exactly four faces coloured in?
<image1>

Answer: 10


Question:
Seven children stand in a circle. Nowhere are two boys found standing next to each other. Nowhere are three girls found standing next to each other. What is possible for the number of girls? The number of girls can...
<image1>

Choices:
- ... only be 3.
- ... be 3 or 4.
- ... be 4 or 5.
- ... only be 5.
- ... only be 4.

logic

Question:
Seven children stand in a circle. Nowhere are two boys found standing next to each other. Nowhere are three girls found standing next to each other. What is possible for the number of girls? The number of girls can...
<image1>
(A) ... only be 3.
(B) ... be 3 or 4.
(C) ... be 4 or 5.
(D) ... only be 5.
(E) ... only be 4.


Question:
A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
<image1>
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
<image2>

Answer: 52

Solution: A solution can be obtained by reflecting the square repeatedly in the cushion the ball strikes. The path of the ball is then represented by the line $A B^{\prime}$ in the diagram. The length of the path can be calculated using Pythagoras Theorem. We have $\left(A B^{\prime}\right)^{2}=(3 \times 2)^{2}+(2 \times 2)^{2}$. Therefore $\left(A B^{\prime}\right)^{2}=36+16=52$ and so $A B^{\prime}=\sqrt{52}$ metres and hence the value of $k$ is 52 . ![](https://cdn.mathpix.com/cropped/2023_12_27_d884587740e101721faeg-3.jpg?height=317&width=491&top_left_y=393&top_left_x=1342)

metric geometry - length

Question:
A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
<image1>
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
<image2>

Answer: 52


Question:
Rosa wants to start at the arrow, follow the line, and get out at the other arrow. Which piece is it NOT possible to put in the middle to obtain that?
<image1>
<image2>

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

combinatorial geometry

Question:
Rosa wants to start at the arrow, follow the line, and get out at the other arrow. Which piece is it NOT possible to put in the middle to obtain that?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Katrin forms a path around each square. For that she uses stones like this
<image1>
How many such stones does she need for a path around the square with side length 5?

Answer: 12

combinatorial geometry

Question:
Katrin forms a path around each square. For that she uses stones like this
<image1>
How many such stones does she need for a path around the square with side length 5?

Answer: 12


Question:
What time is it now, if after 6 hours and 30 minutes the clock will show 4:00?
<image1>

Choices:
- 10:00
- 10:30
- 2:30
- 22:10
- 21:30

arithmetic

Question:
What time is it now, if after 6 hours and 30 minutes the clock will show 4:00?
<image1>
(A) 10:00
(B) 10:30
(C) 2:30
(D) 22:10
(E) 21:30


Answer: 5

logic

Answer: 5


Question:
The distance between the top of the cat that is sitting on the table to the top of the cat that is sleeping on the floor is $150 \mathrm{~cm}$. The distance from the top of the cat that is sleeping on the table to the top of the cat that is sitting on the floor is $110 \mathrm{~cm}$. How high is the table?
<image1>

Choices:
- $110 \mathrm{~cm}$
- $120 \mathrm{~cm}$
- $130 \mathrm{~cm}$
- $140 \mathrm{~cm}$
- $150 \mathrm{~cm}$

algebra

Question:
The distance between the top of the cat that is sitting on the table to the top of the cat that is sleeping on the floor is $150 \mathrm{~cm}$. The distance from the top of the cat that is sleeping on the table to the top of the cat that is sitting on the floor is $110 \mathrm{~cm}$. How high is the table?
<image1>
(A) $110 \mathrm{~cm}$
(B) $120 \mathrm{~cm}$
(C) $130 \mathrm{~cm}$
(D) $140 \mathrm{~cm}$
(E) $150 \mathrm{~cm}$


Question:
The figure shows graphs of functions $f$ and $g$ defined on real numbers. Each graph consists of two perpendicular halflines. Which equality is satisfied for every real number $x$?
<image1>

Choices:
- $f(x)=-g(x)+2$
- $f(x)=-g(x)-2$
- $f(x)=-g(x+2)$
- $f(x+2)=-g(x)$
- $f(x+1)=-g(x-1)$

analytic geometry

Question:
The figure shows graphs of functions $f$ and $g$ defined on real numbers. Each graph consists of two perpendicular halflines. Which equality is satisfied for every real number $x$?
<image1>
(A) $f(x)=-g(x)+2$
(B) $f(x)=-g(x)-2$
(C) $f(x)=-g(x+2)$
(D) $f(x+2)=-g(x)$
(E) $f(x+1)=-g(x-1)$


Question:
A graph consists of 16 points and several connecting lines as shown in the diagram. An ant is at point $A$. With every move the ant can move from the point where it currently is, along one of the connecting lines, to an adjacent point. At which of the points $P, Q, R, S$ and $T$ can the ant be after 2019 moves?
<image1>

Choices:
- only at $P, R$ or $S$, not at $Q$ or $T$
- only at $P$, $R$, $S$ or $T$, not at $Q$
- only at $Q$
- only at $T$
- At all of the points

algebra

Question:
A graph consists of 16 points and several connecting lines as shown in the diagram. An ant is at point $A$. With every move the ant can move from the point where it currently is, along one of the connecting lines, to an adjacent point. At which of the points $P, Q, R, S$ and $T$ can the ant be after 2019 moves?
<image1>
(A) only at $P, R$ or $S$, not at $Q$ or $T$
(B) only at $P$, $R$, $S$ or $T$, not at $Q$
(C) only at $Q$
(D) only at $T$
(E) At all of the points


Question:
Which of the following cubes can be folded from the net on the right?
<image1>
<image2>

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

Solution: When the net is folded up, the two small shaded squares will appear on the same face, and opposite the face consisting of the larger shaded square.

descriptive geometry

Question:
Which of the following cubes can be folded from the net on the right?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: Tom, John and Lily have each shot 6 arrows on a disc with three sections (see diagram). The number of points of a hit depends on the section that has been hit. Tom has 46 points and John has 34 points. How many points did Lily get? <image1>

Answer: 40

algebra

Question: Tom, John and Lily have each shot 6 arrows on a disc with three sections (see diagram). The number of points of a hit depends on the section that has been hit. Tom has 46 points and John has 34 points. How many points did Lily get? <image1>

Answer: 40


Question:
On the number line below, each gap equals one unit. Six integers are marked as shown. At least two of the integers are divisible by 3 , and at least two of them are divisible by 5 . Which of the integers are divisible by 15 ?
<image1>

Choices:
- $F$ and $K$
- $G$ and $J$
- $H$ and $I$
- all six numbers
- only one of them

Solution: To be divisible by three, at least two integers must differ by a multiple of three, which is true for all of $F, G$, $J$, and $K$, so they must all be divisible by 3 . To be divisible by five, there must be at least two integers that differ by a multiple of 5 , which is true for all of $F, H, I$, and $K$, so they must all be divisible by 5 . Hence $F$ and $K$ are divisible by 3 and 5, so are divisible by 15 .

algebra

Question:
On the number line below, each gap equals one unit. Six integers are marked as shown. At least two of the integers are divisible by 3 , and at least two of them are divisible by 5 . Which of the integers are divisible by 15 ?
<image1>
(A) $F$ and $K$
(B) $G$ and $J$
(C) $H$ and $I$
(D) all six numbers
(E) only one of them


Question:
The first diagram on the right shows a shape constructed from two rectangles. The lengths of two sides are marked: 11 and 13. The shape is cut into three parts and the parts are rearranged, as shown in the second diagram on the right. What is the length marked $x$ ?
<image1>

Answer: 37

Solution: The original shape constructed from two rectangles has base of length $11+13=24$. By considering the rearrangement, the lengths 11,13,24 and $x$ can be identified as shown in the diagram. Hence $x=13+24=37$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-132.jpg?height=302&width=382&top_left_y=1625&top_left_x=1428)

metric geometry - length

Question:
The first diagram on the right shows a shape constructed from two rectangles. The lengths of two sides are marked: 11 and 13. The shape is cut into three parts and the parts are rearranged, as shown in the second diagram on the right. What is the length marked $x$ ?
<image1>

Answer: 37


Question: Kristina has a piece of see-through foil on which some points and lines are drawn. She folds the foil along the dotted line. <image1> What can she see now?

Choices:
- $2\vdots6\vdots9$
- $2\vdots6\vdots6$
- $5\vdots6\vdots6$
- $2\vdots8\vdots6$
- $5\vdots8\vdots9$

transformation geometry

Question:
Kristina has a piece of see-through foil on which some points and lines are drawn. She folds the foil along the dotted line. <image1> What can she see now?
(A) $2\vdots6\vdots9$
(B) $2\vdots6\vdots6$
(C) $5\vdots6\vdots6$
(D) $2\vdots8\vdots6$
(E) $5\vdots8\vdots9$


Question:
What is the sum of the 10 angles marked in the picture?
<image1>

Choices:
- $720^{\circ}$
- $600^{\circ}$
- $450^{\circ}$
- $360^{\circ}$
- $300^{\circ}$

metric geometry - angle

Question:
What is the sum of the 10 angles marked in the picture?
<image1>
(A) $720^{\circ}$
(B) $600^{\circ}$
(C) $450^{\circ}$
(D) $360^{\circ}$
(E) $300^{\circ}$


Question:
Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$
<image1>

Choices:
- $\frac{1}{2}\csc{\frac{1}{4}}$
- $2\cos{\frac{1}{2}}$
- $4\sin{\frac{1}{2}}$
- $\csc{\frac{1}{2}}$
- $2\sec{\frac{1}{2}}$

metric geometry - length

Question:
Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$
<image1>
(A) $\frac{1}{2}\csc{\frac{1}{4}}$
(B) $2\cos{\frac{1}{2}}$
(C) $4\sin{\frac{1}{2}}$
(D) $\csc{\frac{1}{2}}$
(E) $2\sec{\frac{1}{2}}$


Question:
Eli drew a board on the floor with nine squares and wrote a number on each of them, starting from 1 and adding 3 units to each new number he wrote, until he filled the board. In the picture, three of the numbers that Eli wrote appear. Which number below can be one of the numbers she wrote in the colored box?
<image1>

Choices:
- 10
- 14
- 17
- 20
- 22

logic

Question:
Eli drew a board on the floor with nine squares and wrote a number on each of them, starting from 1 and adding 3 units to each new number he wrote, until he filled the board. In the picture, three of the numbers that Eli wrote appear. Which number below can be one of the numbers she wrote in the colored box?
<image1>
(A) 10
(B) 14
(C) 17
(D) 20
(E) 22


Question:
Zita walks from left to right and puts the numbers into her basket. Which of the following numbers can be in her basket?
<image1>

Choices:
- 1, 2 and 4
- 2, 3 and 4
- 2, 3 and 5
- 1, 5 and 6
- 1, 2 and 5

graph theory

Question:
Zita walks from left to right and puts the numbers into her basket. Which of the following numbers can be in her basket?
<image1>
(A) 1, 2 and 4
(B) 2, 3 and 4
(C) 2, 3 and 5
(D) 1, 5 and 6
(E) 1, 2 and 5


Question:
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $ 4,5,6,7,8,9 $ in one of the six boxes in this addition problem?

<image1>

Answer: 1047

combinatorics

Question:
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $ 4,5,6,7,8,9 $ in one of the six boxes in this addition problem?

<image1>

Answer: 1047


Question: Some edges of a cube are coloured in red so that each sides of the cube has at least one red edge. What is the minimum number of red edges that the cube has? <image1>

Answer: 3

solid geometry

Question: Some edges of a cube are coloured in red so that each sides of the cube has at least one red edge. What is the minimum number of red edges that the cube has? <image1>

Answer: 3


Question:
Maria puts 4 liters of water in vase I, 3 liters of water in vase II and 4 liters of water in vase III, represented on the side. Seen from the front, these three vases seem to have the same size. Which of the following images can represent the three vases, when seen from above?
<image1>
<image2>

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

solid geometry

Question:
Maria puts 4 liters of water in vase I, 3 liters of water in vase II and 4 liters of water in vase III, represented on the side. Seen from the front, these three vases seem to have the same size. Which of the following images can represent the three vases, when seen from above?
<image1>
<image2>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
Maxi builds towers made up of little $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 2 \mathrm{~cm}$ building blocks as can be seen in the picture.
<image1>
He continues to build his towers in the same way. Finally he uses 28 building blocks for one tower. What is the height of this tower?

Choices:
- $9 \mathrm{~cm}$
- $10 \mathrm{~cm}$
- $11 \mathrm{~cm}$
- $12 \mathrm{~cm}$
- $14 \mathrm{~cm}$

algebra

Question:
Maxi builds towers made up of little $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 2 \mathrm{~cm}$ building blocks as can be seen in the picture.
<image1>
He continues to build his towers in the same way. Finally he uses 28 building blocks for one tower. What is the height of this tower?
(A) $9 \mathrm{~cm}$
(B) $10 \mathrm{~cm}$
(C) $11 \mathrm{~cm}$
(D) $12 \mathrm{~cm}$
(E) $14 \mathrm{~cm}$


Question:
The digits $2$, $0$, $2$, and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

<image1>

Answer: 9

combinatorics

Question:
The digits $2$, $0$, $2$, and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

<image1>

Answer: 9


Question:
Five cards are lying on the table in the order 5, 1, 4, 3, 2. You must get the cards in the order 1, 2, 3, 4, 5. Per move, any two cards may be interchanged. How many moves do you need at least?
<image1>

Answer: 3

algebra

Question:
Five cards are lying on the table in the order 5, 1, 4, 3, 2. You must get the cards in the order 1, 2, 3, 4, 5. Per move, any two cards may be interchanged. How many moves do you need at least?
<image1>

Answer: 3


Question:
An equilateral triangle is being rolled around a unit square as shown. How long is the path that the point shown covers, if the point and the triangle are both back at the start for the first time?
<image1>

Choices:
- $4 \pi$
- $\frac{28}{3} \pi$
- $8 \pi$
- $\frac{14}{3} \pi$
- $\frac{21}{2} \pi$

transformation geometry

Question:
An equilateral triangle is being rolled around a unit square as shown. How long is the path that the point shown covers, if the point and the triangle are both back at the start for the first time?
<image1>
(A) $4 \pi$
(B) $\frac{28}{3} \pi$
(C) $8 \pi$
(D) $\frac{14}{3} \pi$
(E) $\frac{21}{2} \pi$


Question:
Anna has connected all the upper and lower points with straight lines. How many lines has she drawn?
<image1>

Answer: 30

arithmetic

Question:
Anna has connected all the upper and lower points with straight lines. How many lines has she drawn?
<image1>

Answer: 30


Question:
In the picture on the right we see an L-shaped object which is made up of four squares. We would like to add another equally big square so that the new object has a line of symmetry. How many ways are there to achieve this?
<image1>

Answer: 3

combinatorial geometry

Question:
In the picture on the right we see an L-shaped object which is made up of four squares. We would like to add another equally big square so that the new object has a line of symmetry. How many ways are there to achieve this?
<image1>

Answer: 3


Question:
Marc builds the number 2022 as seen in the picture by glueing together 66 cubes of the same size. Afterwards he paints the entire surface of his work. On how many of the 66 cubes has Marc painted exactly four faces?
<image1>

Answer: 60

solid geometry

Question:
Marc builds the number 2022 as seen in the picture by glueing together 66 cubes of the same size. Afterwards he paints the entire surface of his work. On how many of the 66 cubes has Marc painted exactly four faces?
<image1>

Answer: 60


Question:
In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure.  The missing piece (the mouth) has central angle $ 60^{\circ}$.  What is the perimeter of the monster in cm?

<image1>

Choices:
- $\pi + 2$
- $2\pi$
- $\frac{5}{3} \pi$
- $\frac{5}{6} \pi + 2$
- $\frac{5}{3} \pi + 2$

metric geometry - length

Question:
In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure.  The missing piece (the mouth) has central angle $ 60^{\circ}$.  What is the perimeter of the monster in cm?

<image1>
(A) $\pi + 2$
(B) $2\pi$
(C) $\frac{5}{3} \pi$
(D) $\frac{5}{6} \pi + 2$
(E) $\frac{5}{3} \pi + 2$


Question:
Meike paddles around five buoys with her boat (see diagram). Which of the buoys does she paddle around in a clockwise direction?
<image1>

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

topology

Question:
Meike paddles around five buoys with her boat (see diagram). Which of the buoys does she paddle around in a clockwise direction?
<image1>
(A) 2, 3 and 4
(B) 1, 2 and 3
(C) 1, 3 and 5
(D) 2, 4 and 5
(E) 2, 3 and 5


Question:
A large set of weighing scales has two identical sets of scales placed on it, one on each pan. Four weights $W, X, Y, Z$ are placed on the weighing scales as shown in the left diagram.
<image1>
Then two of these weights are swapped, and the pans now appear as shown in the diagram on the right. Which two weights were swapped?

Choices:
- $W$ and $Z$
- $W$ and $Y$
- $W$ and $X$
- $X$ and $Z$
- $X$ and $Y$

Solution: From the first picture, we can see:  \begin{tabular}{lrl}  From the right scale: & $Z$ & $>Y$ \\ From the left scale: & $X$ & $>W$ \\ From the large scale: & $Y+Z>W+X$ \end{tabular}  It follows from (1), (2) and (3) that $Z+Z>Y+Z>W+X>W+W$, so $2 Z>2 W$, and hence $Z>W \ldots(4)$.  We can show that most swaps give a contradiction of these inequalities:  Firstly, suppose that weight $Z$ doesn't move. Then there are three possible swaps: $X$ and $Y$ swap: then in the second picture we must have $Z<X$ and $Y<W$, which add to give $Y+Z<W+X$, contradicting (3).  $Y$ and $W$ swap: then we would have $Z<W$, contradicting (4).  $X$ and $W$ swap: then we would have $X<W$, contradicting (2).  Hence $Z$ must swap, which can happen in three ways:  $Z$ and $Y$ swap: then we would have $Z<Y$, contradicting (1).  $Z$ and $X$ swap: then we would have $Z<W$, contradicting (4).  $Z$ and $W$ swap: This is the only possibility left, and it can work when the weights $W, X, Y, Z$ are 1, 2, 3, 4. The swap would give 4, 2, 3, 1 which matches the picture on the right. However, this does depend on the values of $W, X, Y, Z$; it would not work for 1, 5, 3, 4 .

algebra

Question:
A large set of weighing scales has two identical sets of scales placed on it, one on each pan. Four weights $W, X, Y, Z$ are placed on the weighing scales as shown in the left diagram.
<image1>
Then two of these weights are swapped, and the pans now appear as shown in the diagram on the right. Which two weights were swapped?
(A) $W$ and $Z$
(B) $W$ and $Y$
(C) $W$ and $X$
(D) $X$ and $Z$
(E) $X$ and $Y$


Question:
Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
<image1>

Choices:
- $\pi-\sqrt{3}$
- $\pi-\sqrt{2}$
- $\frac{\pi+\sqrt{2}}{2}$
- $\frac{\pi+\sqrt{3}}{2}$
- $\frac{7}{6}\pi-\frac{\sqrt{3}}{2}$

metric geometry - area

Question:
Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
<image1>
(A) $\pi-\sqrt{3}$
(B) $\pi-\sqrt{2}$
(C) $\frac{\pi+\sqrt{2}}{2}$
(D) $\frac{\pi+\sqrt{3}}{2}$
(E) $\frac{7}{6}\pi-\frac{\sqrt{3}}{2}$


Question:
The diagram shows a square $P Q R S$ with side length 1. The point $U$ is the midpoint of the side $R S$ and the point $W$ is the midpoint of the square. The three line segments, $T W, U W$ and $V W$ split the square into three equally big areas. How long is the line segment $S V$?
<image1>

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

metric geometry - length

Question:
The diagram shows a square $P Q R S$ with side length 1. The point $U$ is the midpoint of the side $R S$ and the point $W$ is the midpoint of the square. The three line segments, $T W, U W$ and $V W$ split the square into three equally big areas. How long is the line segment $S V$?
<image1>
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{4}{5}$
(E) $\frac{5}{6}$


Question:
Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
<image1>

Choices:
- 48
- 75
- 151
- 192
- 603

solid geometry

Question:
Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
<image1>
(A) 48
(B) 75
(C) 151
(D) 192
(E) 603


Question:
The diagram shows a triangle $A B C$ with area $12 \mathrm{~cm}^{2}$. The sides of the triangle are extended to points $P, Q, R, S, T$ and $U$ as shown so that $P A=A B=B S, Q A=A C=C T$ and $R B=B C=C U$.
<image1>
What is the area (in $\mathrm{cm}^{2}$ ) of hexagon $P Q R S T U$ ?

Answer: 156

Solution: Consider triangles $A B C$ and $A S T$. Angles $C A B$ and $T A S$ are equal because they are the same angle, $S A=2 B A$ and $T A=2 C A$. Hence triangles $A B C$ and $A S T$ are similar. The ratio of their sides is $1: 2$ and hence the ratio of their areas is $1^{2}: 2^{2}=1: 4$. Therefore the area of triangle $A S T$ is $4 \times 12 \mathrm{~cm}^{2}=48 \mathrm{~cm}^{2}$ and hence the area of $B S T C$ is $(48-12) \mathrm{cm}^{2}=36 \mathrm{~cm}^{2}$. In a similar way, it ![](https://cdn.mathpix.com/cropped/2023_12_27_d884587740e101721faeg-2.jpg?height=372&width=562&top_left_y=1096&top_left_x=1341) can be shown that each of the areas of $C U P A$ and $A Q R B$ is also $36 \mathrm{~cm}^{2}$. Next consider triangles $A B C$ and $A P Q$. Angles $B A C$ and $P A Q$ are equal using vertically opposite angles, $A B=A P$ and $A C=A Q$. Hence triangles $A B C$ and $A P Q$ are congruent (SAS) and so the area of triangle $A P Q$ is $12 \mathrm{~cm}^{2}$. In a similar way, it can be shown that the each of areas of triangles $B R S$ and $C T U$ is also $12 \mathrm{~cm}^{2}$. Hence the total area of hexagon PQRSTU in $\mathrm{cm}^{2}$ is $(3 \times 36+4 \times 12)=156$.

metric geometry - area

Question:
The diagram shows a triangle $A B C$ with area $12 \mathrm{~cm}^{2}$. The sides of the triangle are extended to points $P, Q, R, S, T$ and $U$ as shown so that $P A=A B=B S, Q A=A C=C T$ and $R B=B C=C U$.
<image1>
What is the area (in $\mathrm{cm}^{2}$ ) of hexagon $P Q R S T U$ ?

Answer: 156


Question:
A rectangle $A B C D$ with dimensions $16 \mathrm{~cm}$ by $4 \mathrm{~cm}$ was folded along the line MN so that corner C meets corner A. What is the area of the Pentagon ABNMD'?
<image1>

Choices:
- $17 \mathrm{~cm}^{2}$
- $27 \mathrm{~cm}^{2}$
- $37 \mathrm{~cm}^{2}$
- $47 \mathrm{~cm}^{2}$
- $57 \mathrm{~cm}^{2}$

transformation geometry

Question:
A rectangle $A B C D$ with dimensions $16 \mathrm{~cm}$ by $4 \mathrm{~cm}$ was folded along the line MN so that corner C meets corner A. What is the area of the Pentagon ABNMD'?
<image1>
(A) $17 \mathrm{~cm}^{2}$
(B) $27 \mathrm{~cm}^{2}$
(C) $37 \mathrm{~cm}^{2}$
(D) $47 \mathrm{~cm}^{2}$
(E) $57 \mathrm{~cm}^{2}$


Question:
Julia has 5 pieces of plastic and has stacked these pieces on a table, as shown beside. What was the second piece she put on the table?
<image1>
<image2>

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

descriptive geometry

Question:
Julia has 5 pieces of plastic and has stacked these pieces on a table, as shown beside. What was the second piece she put on the table?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The figure should be rotated $180^{\circ}$ around point $\mathrm{F}$. What is the result?
<image1>
<image2>

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

transformation geometry

Question:
The figure should be rotated $180^{\circ}$ around point $\mathrm{F}$. What is the result?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The diagram shows a square and an equilateral right-angled crossshaped dodecagon. The length of the perimeter of the dodecagon is $36 \mathrm{~cm}$. What is the area of the square in $\mathrm{cm}^{2}$ ? <image1>

Answer: 72

Solution: Each side of the dodecagon is of length $36 \div 12 \mathrm{~cm}=3 \mathrm{~cm}$. Calculating the marked length gives $x=\sqrt{18}$. The square has side of length $2 \sqrt{18} \mathrm{~cm}$ and area $(2 \sqrt{18})^{2} \mathrm{~cm}^{2}=72 \mathrm{~cm}^{2}$.  $O R$ The dodecagon can be divided into 5 squares of area $9 \mathrm{~cm}^{2}$. The white triangles can be reassembled into 3 equal squares making 8 in total. So the total area is $8 \times 9 \mathrm{~cm}^{2}=72 \mathrm{~cm}^{2}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-031.jpg?height=209&width=211&top_left_y=2431&top_left_x=1636)

metric geometry - area

Question: The diagram shows a square and an equilateral right-angled crossshaped dodecagon. The length of the perimeter of the dodecagon is $36 \mathrm{~cm}$. What is the area of the square in $\mathrm{cm}^{2}$ ? <image1>

Answer: 72


Question:
Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray-and-white tiles shown below on the right.<image1>
What is the probability that the tiling will contain a large gray diamond in one of the smaller $2\times 2$ grids? Below is an example of one such tiling.
<image2>

Choices:
- $\frac{1}{1024}$
- $\frac{1}{256}$
- $\frac{1}{64}$
- $\frac{1}{16}$
- $\frac{1}{4}$

combinatorial geometry

Question:
Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray-and-white tiles shown below on the right.<image1>
What is the probability that the tiling will contain a large gray diamond in one of the smaller $2\times 2$ grids? Below is an example of one such tiling.
<image2>
(A) $\frac{1}{1024}$
(B) $\frac{1}{256}$
(C) $\frac{1}{64}$
(D) $\frac{1}{16}$
(E) $\frac{1}{4}$


Question:
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
<image1>

Answer: 6

combinatorial geometry

Question:
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
<image1>

Answer: 6


Question:
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued.  The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each.  (Brazil and Peru are South American countries and France and Spain are in Europe.)
<image1>

The average price of his '70s stamps is closest to

Choices:
- $3.5 \text{ cents}$
- $4 \text{ cents}$
- $4.5 \text{ cents}$
- $5 \text{ cents}$
- $5.5 \text{ cents}$

statistics

Question:
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued.  The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each.  (Brazil and Peru are South American countries and France and Spain are in Europe.)
<image1>

The average price of his '70s stamps is closest to
(A) $3.5 \text{ cents}$
(B) $4 \text{ cents}$
(C) $4.5 \text{ cents}$
(D) $5 \text{ cents}$
(E) $5.5 \text{ cents}$


Question:
How many fish will have their heads pointing towards the ring when we straighten the line?
<image1>

Answer: 6

topology

Question:
How many fish will have their heads pointing towards the ring when we straighten the line?
<image1>

Answer: 6


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:
Ten people order an ice cream for each one. They ask for four vanilla ice creams, three chocolate ice creams, two lemon ice creams and one mango ice cream. As a topping, they ask for four umbrellas, three cherries, two wafers and a chocolate gum, one topping for each ice cream. Since they don't want two ice creams the same, which of the following combinations is possible?
<image1>

Choices:
- Chocolat and chocolate gum.
- Mango and cherry.
- Lemmon and wafer.
- Mango and wafer.
- Lemmon and cherry.

logic

Question:
Ten people order an ice cream for each one. They ask for four vanilla ice creams, three chocolate ice creams, two lemon ice creams and one mango ice cream. As a topping, they ask for four umbrellas, three cherries, two wafers and a chocolate gum, one topping for each ice cream. Since they don't want two ice creams the same, which of the following combinations is possible?
<image1>
(A) Chocolat and chocolate gum.
(B) Mango and cherry.
(C) Lemmon and wafer.
(D) Mango and wafer.
(E) Lemmon and cherry.


Question:
A fifteen-meter log has to be sawn into three-meter pieces. How many cuts are needed for that?
<image1>

Answer: 4

arithmetic

Question:
A fifteen-meter log has to be sawn into three-meter pieces. How many cuts are needed for that?
<image1>

Answer: 4


Question:
The dartboard shown consists of an inner black circle and 2 rings around it. The width of each ring is equal to the radius of the black circle. How many times greater is the area of the grey ring than the area of the inner black circle?
<image1>

Answer: 5

metric geometry - area

Question:
The dartboard shown consists of an inner black circle and 2 rings around it. The width of each ring is equal to the radius of the black circle. How many times greater is the area of the grey ring than the area of the inner black circle?
<image1>

Answer: 5


Question:
$ABCD$ is a square 4 inches on a side, and each of the inside squares is formed by joining the midpoints of the outer square's sides. What is the area of the shaded region in square inches?

<image1>

Answer: 4

Solution: Connecting the midpoints of the sides of a square in order forms a square that has half the area of the original square.  So, the square formed by connecting the midpoints of $ABCD$ has area $\frac{1}{2}\cdot 4^2 = 8$ square inches.  The smallest square in the diagram is formed by connecting the midpoints of this square with area 8, so the smallest square has area $\frac{1}{2} \cdot 8 =4$ square inches, leaving $8-4=\boxed{4}$ square inches of shaded area.

metric geometry - area

Question:
$ABCD$ is a square 4 inches on a side, and each of the inside squares is formed by joining the midpoints of the outer square's sides. What is the area of the shaded region in square inches?

<image1>

Answer: 4


Question:
Which piece is missing?
<image1>
<image2>

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

transformation geometry

Question:
Which piece is missing?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
One of the two sides of a rectangle has length $6 \mathrm{~cm}$. In the rectangle circles are drawn next to each other in such a way that their centres form an equilateral triangle. What is the shortest distance between the two grey circles (in $\mathrm{cm}$ )?
<image1>

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

metric geometry - length

Question:
One of the two sides of a rectangle has length $6 \mathrm{~cm}$. In the rectangle circles are drawn next to each other in such a way that their centres form an equilateral triangle. What is the shortest distance between the two grey circles (in $\mathrm{cm}$ )?
<image1>
(A) 1
(B) $\sqrt{2}$
(C) $2 \sqrt{3}-2$
(D) $\frac{\pi}{2}$
(E) 2


Question:
The solid in the diagram is made out of 8 identical cubes. What does the solid look like when viewed from above?
<image1>
<image2>

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

descriptive geometry

Question:
The solid in the diagram is made out of 8 identical cubes. What does the solid look like when viewed from above?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Hansi sticks 12 cubes together to make this figure. He always puts one drop of glue between two cubes. How many drops of glue does he need?
<image1>

Answer: 11

solid geometry

Question:
Hansi sticks 12 cubes together to make this figure. He always puts one drop of glue between two cubes. How many drops of glue does he need?
<image1>

Answer: 11


Question:
On Nadya's smartphone, the diagram shows how much time she spent last week on four of her apps. This week she halved the time spent on two of these apps, but spent the same amount of time as the previous week on the other two apps.
<image1>
Which of the following could be the diagram for this week?
<image2>

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

Solution: In Diagram $\mathrm{E}$ the times for the first and third apps have been halved, while the other two are unchanged. It can be easily checked that the other diagrams do not work.

statistics

Question:
On Nadya's smartphone, the diagram shows how much time she spent last week on four of her apps. This week she halved the time spent on two of these apps, but spent the same amount of time as the previous week on the other two apps.
<image1>
Which of the following could be the diagram for this week?
<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:
A machine-shop cutting tool has the shape of a notched circle, as shown.  The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm.  The angle $ABC$ is a right angle.  Find the square of the distance (in centimeters) from $B$ to the center of the circle.
<image1>

Answer: 26

metric geometry - length

Question:
A machine-shop cutting tool has the shape of a notched circle, as shown.  The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm.  The angle $ABC$ is a right angle.  Find the square of the distance (in centimeters) from $B$ to the center of the circle.
<image1>

Answer: 26


Question:
John has a chocolate tablet consisting of square pieces of $1 \mathrm{~cm} \times 1 \mathrm{~cm}$. He has eaten already some pieces in a corner (see the picture). How many pieces John still have?
<image1>

Answer: 60

metric geometry - area

Question:
John has a chocolate tablet consisting of square pieces of $1 \mathrm{~cm} \times 1 \mathrm{~cm}$. He has eaten already some pieces in a corner (see the picture). How many pieces John still have?
<image1>

Answer: 60


Question:
A large rectangular plot is divided into two lots that are separated from each other by an $A B C D$ fence, as shown in the picture beside. The $A B, B C$ and $C D$ parts of this fence are parallel to the sides of the rectangle and have lengths of $30 \mathrm{~m}$, $24 \mathrm{~m}$ and $10 \mathrm{~m}$, respectively. The owners of these lands have combined to knock down the fence and make a new straight AE fence, without changing the area of each of the lands. How far from point $D$ should the $E$ end of the fence be?
<image1>

Choices:
- $8 \mathrm{~m}$
- $10 \mathrm{~m}$
- $12 \mathrm{~m}$
- $14 \mathrm{~m}$
- $16 \mathrm{~m}$

metric geometry - length

Question:
A large rectangular plot is divided into two lots that are separated from each other by an $A B C D$ fence, as shown in the picture beside. The $A B, B C$ and $C D$ parts of this fence are parallel to the sides of the rectangle and have lengths of $30 \mathrm{~m}$, $24 \mathrm{~m}$ and $10 \mathrm{~m}$, respectively. The owners of these lands have combined to knock down the fence and make a new straight AE fence, without changing the area of each of the lands. How far from point $D$ should the $E$ end of the fence be?
<image1>
(A) $8 \mathrm{~m}$
(B) $10 \mathrm{~m}$
(C) $12 \mathrm{~m}$
(D) $14 \mathrm{~m}$
(E) $16 \mathrm{~m}$


Question:
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y + z$.
<image1>

Answer: 46

algebra

Question:
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y + z$.
<image1>

Answer: 46


Question:
Two squares have side 1. What is the area of the black quadrangle?
<image1>

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

metric geometry - area

Question:
Two squares have side 1. What is the area of the black quadrangle?
<image1>
(A) $\sqrt{2}-1$
(B) $\frac{\sqrt{2}}{2}$
(C) $\frac{\sqrt{2}+1}{2}$
(D) $\sqrt{2}+1$
(E) $\sqrt{3}-\sqrt{2}$


Question:
Diana shoots 3 darts, three times at a target board with two fields. The first time she scores 12 points, the second time 15. The number of points depends on which field she has hit. How many points does she score the third time?
<image1>

Answer: 21

algebra

Question:
Diana shoots 3 darts, three times at a target board with two fields. The first time she scores 12 points, the second time 15. The number of points depends on which field she has hit. How many points does she score the third time?
<image1>

Answer: 21


Question:
In the figure shown there are three concentric circles and two perpendicular diameters. The three shaded regions have equal area. The radius of the small circle is 2 . The product of the three radii is $Y$.
What is the value of $Y^{2}$ ?
<image1>

Answer: 384

Solution: We note that the central circle, the inner hoop and the outer hoop are also equal in area. Let the radii of the middle circle and outer circle be $a$ and $b$ respectively. The area of the central circle is $4 \pi$. The area of the inner hoop is $\pi a^{2}-4 \pi$. Therefore, $\pi a^{2}-4 \pi=4 \pi$. This simplifies to $a^{2}=8$. The area of the outer hoop is $\pi b^{2}-\pi a^{2}=\pi b^{2}-8 \pi$. Therefore, $\pi b^{2}-8 \pi=4 \pi$. This simplifies to $b^{2}=12$. The product of the radii, $Y$, is $2 \times a \times b$. Therefore, $Y^{2}=4 \times a^{2} \times b^{2}=4 \times 8 \times 12=384$.

metric geometry - length

Question:
In the figure shown there are three concentric circles and two perpendicular diameters. The three shaded regions have equal area. The radius of the small circle is 2 . The product of the three radii is $Y$.
What is the value of $Y^{2}$ ?
<image1>

Answer: 384


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:
Carl writes down a five-digit number.
He then places a shape on each of the five digits (see picture).
He places different shapes on different digits.
He places the same shape on the same digits.
Which number did Carl hide?
<image1>

Answer: 34426

logic

Question:
Carl writes down a five-digit number.
He then places a shape on each of the five digits (see picture).
He places different shapes on different digits.
He places the same shape on the same digits.
Which number did Carl hide?
<image1>

Answer: 34426


Question: Roo has a very unusual chessboard of side 7, in which only the squares which lie on the diagonals are shaded. Kanga then asks the question "What would be the total white area of your chessboard if each side was 2003 squares long?" What is the correct answer? <image1>

Choices:
- $2002^{2}$
- $2002 \times 2001$
- $2003^{2}$
- $2003 \times 2004$
- $2004^{2}$

Solution: In the large board there would be $2003+2002$ shaded squares. If instead this number of squares were shaded along two adjacent edges, the white region would then be a square of side 2002 .

metric geometry - area

Question:
Roo has a very unusual chessboard of side 7, in which only the squares which lie on the diagonals are shaded. Kanga then asks the question "What would be the total white area of your chessboard if each side was 2003 squares long?" What is the correct answer? <image1>
(A) $2002^{2}$
(B) $2002 \times 2001$
(C) $2003^{2}$
(D) $2003 \times 2004$
(E) $2004^{2}$


Question:
Let $a$ and $b$ be two shorter sides of the right-angled triangle. Then the sum of the diameter of the incircle and that of the circumcircle of this triangle is equal to:
<image1>

Choices:
- $\sqrt{a^{2}+b^{2}}$
- $\sqrt{a b}$
- $0.5(a+b)$
- $2(a+b)$
- $a+b$

metric geometry - length

Question:
Let $a$ and $b$ be two shorter sides of the right-angled triangle. Then the sum of the diameter of the incircle and that of the circumcircle of this triangle is equal to:
<image1>
(A) $\sqrt{a^{2}+b^{2}}$
(B) $\sqrt{a b}$
(C) $0.5(a+b)$
(D) $2(a+b)$
(E) $a+b$


Question:
Which of the following pieces do I need to complete the cuboid?
<image1>
<image2>

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

combinatorial geometry

Question:
Which of the following pieces do I need to complete the cuboid?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle with sides $6 \mathrm{~cm}$ as shown. The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining hexagon. What is the side-length of one of the small triangles? <image1>

Choices:
- $1 \mathrm{~cm}$
- $1.2 \mathrm{~cm}$
- $1.25 \mathrm{~cm}$
- $1.5 \mathrm{~cm}$
- $2 \mathrm{~cm}$

Solution: If we let the length of the side of one of the removed triangles be $x \mathrm{~cm}$, the perimeter of the remaining hexagon will be $3 x+3(6-2 x) \mathrm{cm}$. Hence we have $3(3 x)=3 x+3(6-2 x)$ which has solution $x=18 / 12=1.5$.

metric geometry - length

Question:
Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle with sides $6 \mathrm{~cm}$ as shown. The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining hexagon. What is the side-length of one of the small triangles? <image1>
(A) $1 \mathrm{~cm}$
(B) $1.2 \mathrm{~cm}$
(C) $1.25 \mathrm{~cm}$
(D) $1.5 \mathrm{~cm}$
(E) $2 \mathrm{~cm}$


Question:
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cub is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
<image1>

<image2>

Choices:
- 6$
- $7$
- $\frac{419}{51}$
- $\frac{158}{17}$
- $11$

solid geometry

Question:
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cub is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
<image1>

<image2>
(A) 6$
(B) $7$
(C) $\frac{419}{51}$
(D) $\frac{158}{17}$
(E) $11$


Question:
Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. Paul laid six dominoes in a line as shown in the diagram.
<image1>
He can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one domino. What is the smallest number of moves he needs to make to arrange all the dominoes correctly?

Answer: 3

Solution: The dominoes in the line contain three ends with four spots and three ends with six spots, as shown in diagram 1. Therefore, a correctly arranged set of these dominoes will have four spots at one end and six spots at the other, as is currently the case. Hence, Paul does not need to move either of the end dominoes.  If he swaps the third and the fifth dominoes from diagram 1, he obtains the row shown in diagram 2 which has the same number of spots in the adjacent ends of the fourth, fifth and sixth dominoes. Next, if he swaps the second and third dominoes from diagram 2 to obtain the line shown in diagram 3 , he has matched the spots at the adjacent ends of the first and second dominoes. Finally, rotating the third domino in diagram 3, he obtains the correctly arranged line as shown in diagram 4 . This shows that it is possible to arrange the dominos correctly in three moves.  To see that two moves is not sufficient, note that, whatever else needs to happen, the two $1 \mathrm{~s}$ must be correctly placed next to each other. To do that requires one of the dominos with a 1 to be rotated and then one pair of dominos to be swapped so that the two $1 \mathrm{~s}$ are now next to each other. A similar argument applies to the two dominos with a 3. However, this is not possible in only two moves. Therefore the smallest number of moves he needs to make is 3 .  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-246.jpg?height=362&width=1542&top_left_y=458&top_left_x=400)

combinatorics

Question:
Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. Paul laid six dominoes in a line as shown in the diagram.
<image1>
He can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one domino. What is the smallest number of moves he needs to make to arrange all the dominoes correctly?

Answer: 3


Question:
Karin wants to place five bowls on a table so that they are ordered according to their weight. She has already placed the bowls $Q, R, S$ and $T$ in order, where $Q$ is lightest and $T$ is heaviest. Where does she have to place bowl Z?
<image1>

Choices:
- to the left of bowl Q
- between bowls Q and R
- between bowls R and S
- between bowls S and T
- to the right of bowl T

algebra

Question:
Karin wants to place five bowls on a table so that they are ordered according to their weight. She has already placed the bowls $Q, R, S$ and $T$ in order, where $Q$ is lightest and $T$ is heaviest. Where does she have to place bowl Z?
<image1>
(A) to the left of bowl Q
(B) between bowls Q and R
(C) between bowls R and S
(D) between bowls S and T
(E) to the right of bowl T


Question: Four cars enter a roundabout at the same time, each one from a different direction, as shown in the diagram. Each car drives in a clockwise direction and leaves the roundabout before making a complete circuit. No two cars leave the roundabout by the same exit. How many different ways are there for the cars to leave the roundabout? <image1>

Answer: 9

Solution: Label the cars $1,2,3$ and 4 and their original junctions $P, Q, R$ and $S$ respectively. Whichever junction car 1 leaves by, any of the other three cars could leave by junction $P$. Once car 1 and the car leaving by junction $P$ have been assigned their junctions, we have to consider the other two cars and the other two junctions. However, at least one of these remaining junctions will be the original junction of one of these two cars. Therefore there will be exactly one way in which the two remaining cars can leave by the two remaining junctions. So car 1 can leave by one of three junctions and, for each of these, the remaining cars can leave the roundabout in three different ways. Hence the total number of ways the cars can leave the roundabout is $3 \times 3=9$.

combinatorics

Question: Four cars enter a roundabout at the same time, each one from a different direction, as shown in the diagram. Each car drives in a clockwise direction and leaves the roundabout before making a complete circuit. No two cars leave the roundabout by the same exit. How many different ways are there for the cars to leave the roundabout? <image1>

Answer: 9


Question:
A regular octagon is folded three times down the middle as shown, until a triangle is formed. Then the rightmost corner is cut away. Which of the following shapes is formed when the paper is unfolded?
<image1>
<image2>

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

transformation geometry

Question:
A regular octagon is folded three times down the middle as shown, until a triangle is formed. Then the rightmost corner is cut away. Which of the following shapes is formed when the paper is unfolded?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The points $P$ and $Q$ are opposite vertices of a regular hexagon and the points $R$ and $S$ are midpoints of opposite edges, as shown. The area of the hexagon is $60 \mathrm{~cm}^{2}$. What is the product of the lengths, in cms, of $P Q$ and $R S$ ? <image1>

Answer: 80

Solution: The hexagon can be split into six congruent equilateral triangles. Each triangle has base of length $\frac{1}{2} P Q$ and height $\frac{1}{2} R S$, so the total area is  $$ 6 \times \frac{1}{2} \times\left(\frac{1}{2} P Q\right) \times\left(\frac{1}{2} R S\right)=\frac{3}{4} \times P Q \times R S=60 \mathrm{~cm}^{2} $$  Hence $P Q \times R S=\frac{4}{3} \times 60=80$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-171.jpg?height=366&width=394&top_left_y=294&top_left_x=1459)

metric geometry - length

Question: The points $P$ and $Q$ are opposite vertices of a regular hexagon and the points $R$ and $S$ are midpoints of opposite edges, as shown. The area of the hexagon is $60 \mathrm{~cm}^{2}$. What is the product of the lengths, in cms, of $P Q$ and $R S$ ? <image1>

Answer: 80


Question:
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?

<image1>

Answer: 4

algebra

Question:
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?

<image1>

Answer: 4


Question: $7$ congruent squares are arranged into a 'C,' as shown below. If the perimeter and area of the 'C' are equal (ignoring units), compute the (nonzero) side length of the squares.\n<image1>

Answer: $\boxed{\frac{16}{7}}$

Solution: Let the side length of each of the squares be $x.$\nThen, we can easily find that the perimeter of the figure is $16x$ and the area is $7x^2,$ so\n$$\n\begin{align*}\n16x=7x^2 &\implies 16=7x \\n&\implies x = \boxed{\frac{16}{7}}.\n\end{align*}\n$$

metric geometry - length

Question: $7$ congruent squares are arranged into a 'C,' as shown below. If the perimeter and area of the 'C' are equal (ignoring units), compute the (nonzero) side length of the squares.\n<image1>

Answer: $\boxed{\frac{16}{7}}$


Question:
A rectangle is split into 11 smaller rectangles as shown. All 11 small rectangles are similar to the initial rectangle. The smallest rectangles are aligned like the original rectangle (see diagram). The lower sides of the smallest rectangles have length 1. How big is the perimeter of the big rectangle?
<image1>

Answer: 30

metric geometry - length

Question:
A rectangle is split into 11 smaller rectangles as shown. All 11 small rectangles are similar to the initial rectangle. The smallest rectangles are aligned like the original rectangle (see diagram). The lower sides of the smallest rectangles have length 1. How big is the perimeter of the big rectangle?
<image1>

Answer: 30


Question:
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point.  The areas of four of these triangles are as indicated.  Find the area of triangle $ABC$.

<image1>

Answer: 315

metric geometry - area

Question:
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point.  The areas of four of these triangles are as indicated.  Find the area of triangle $ABC$.

<image1>

Answer: 315


Question:
The kangaroo is inside how many circles?
<image1>

Answer: 3

counting

Question:
The kangaroo is inside how many circles?
<image1>

Answer: 3


Question:
In $ \triangle ABC$, $ AB = BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE = 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$?
<image1>

Choices:
- $16$
- $\frac{50}{3}$
- $10\sqrt{3}$
- $8\sqrt{5}$
- $18$

metric geometry - area

Question:
In $ \triangle ABC$, $ AB = BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE = 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$?
<image1>
(A) $16$
(B) $\frac{50}{3}$
(C) $10\sqrt{3}$
(D) $8\sqrt{5}$
(E) $18$


Question:
Which picture shows a single large loop?
<image1>

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

topology

Question:
Which picture shows a single large loop?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Maria wants to write whole numbers in the squares of the figure, so that the sum of the numbers in three consecutive squares is always 10. She has already written a number. What number should she write on the gray square?
<image1>

Answer: 2

logic

Question:
Maria wants to write whole numbers in the squares of the figure, so that the sum of the numbers in three consecutive squares is always 10. She has already written a number. What number should she write on the gray square?
<image1>

Answer: 2


Question:
The diagram on the right shows a solid made up of 6 triangles. Each vertex is assigned a number, two of which are indicated. The total of the three numbers on each triangle is the same. What is the total of all five numbers?
<image1>

Answer: 17

solid geometry

Question:
The diagram on the right shows a solid made up of 6 triangles. Each vertex is assigned a number, two of which are indicated. The total of the three numbers on each triangle is the same. What is the total of all five numbers?
<image1>

Answer: 17


Question:
My umbrella has KANGAROO written on top as shown in the diagram. Which one of the following pictures also shows my umbrella?
<image1>
<image2>

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

Solution: In diagrams $A, C$ and $D$, the letters ' $N$ ', ' $R$ ' and ' $G$ ' respectively have been reversed. In diagram $\mathrm{B}$, the letters are not in the order they appear on the original umbrella. Hence only option E shows part of the original umbrella.  (This is immediately clear if you turn the question paper round so the handles are pointing up rather than down.)

descriptive geometry

Question:
My umbrella has KANGAROO written on top as shown in the diagram. Which one of the following pictures also shows my umbrella?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Which figure can be made from the 2 pieces shown on the right?
<image1>
<image2>

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

combinatorial geometry

Question:
Which figure can be made from the 2 pieces shown on the right?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued.  The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each.  (Brazil and Peru are South American countries and France and Spain are in Europe.)
<image1>
In dollars and cents, how much did his South American stampes issued before the '70s cost him?

Choices:
- $\textdollar 0.40$
- $\textdollar 1.06$
- $\textdollar 1.80$
- $\textdollar 2.38$
- $\textdollar 2.64$

statistics

Question:
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued.  The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each.  (Brazil and Peru are South American countries and France and Spain are in Europe.)
<image1>
In dollars and cents, how much did his South American stampes issued before the '70s cost him?
(A) $\textdollar 0.40$
(B) $\textdollar 1.06$
(C) $\textdollar 1.80$
(D) $\textdollar 2.38$
(E) $\textdollar 2.64$


Question:
Which picture shows a single large loop?
<image1>

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

topology

Question:
Which picture shows a single large loop?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Nathalie wanted to build a large cube out of lots of small cubes, just like in picture 1. How many cubes are missing from picture 2 that would be needed to build the large cube?
<image1>

Answer: 7

solid geometry

Question:
Nathalie wanted to build a large cube out of lots of small cubes, just like in picture 1. How many cubes are missing from picture 2 that would be needed to build the large cube?
<image1>

Answer: 7


Question:
A paperstrip is folded three times in the middle. It is then opened again and looked at from the side so that one can see all 7 folds from the side at the same time. Which of the following views is not a possible result?
<image1>

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

transformation geometry

Question:
A paperstrip is folded three times in the middle. It is then opened again and looked at from the side so that one can see all 7 folds from the side at the same time. Which of the following views is not a possible result?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E