Question:
Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
<image1>
On the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively.
We know that $c=b e, a+b=d$ and $e-d=a$.
Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card?

Answer: 103

Solution: Since $e-d=a$ we have $e=a+d$. Therefore, $e>a$ and $e>d$. Since $c=b e$ we have $c>e$ and $c>b$, for if either of $e, b$ equalled 1 then $c$ would equal the other. Therefore, $c>e>a, c>e>d$ and $c>b$. The largest integer is $c$ and the correct answer is 103.

algebra

Question:
Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
<image1>
On the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively.
We know that $c=b e, a+b=d$ and $e-d=a$.
Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card?

Answer: 103


Question:
On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101, 102, 103, 104, 105 and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
<image1>

Choices:
- $75 \%$
- $80 \%$
- $90 \%$
- $95 \%$
- $100 \%$

Solution: The total mass is $621 \mathrm{~g}$ so any three masses with total mass over $310.5 \mathrm{~g}$ could be in the heavier pan. There are eight of these triples that include the $106 \mathrm{~g}$ mass: $(106,105,104)$, $(106,105,103),(106,105,102),(106,105,101),(106,104,103),(106,104,102),(106$, $104,101)$, and $(106,103,102)$.  Without 106, there are 2 ways to make a set over $310.5 \mathrm{~g}:(105,104,103)$ and $(105,104$, 102).  Hence the probabililty that the $106 \mathrm{~g}$ mass is included in the heavier pan is $\frac{8}{8+2}=\frac{8}{10}$ or $80 \%$.

combinatorics

Question:
On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101, 102, 103, 104, 105 and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
<image1>
(A) $75 \%$
(B) $80 \%$
(C) $90 \%$
(D) $95 \%$
(E) $100 \%$


Question:
Two angles are marked on the $3 \times 3$ grid of squares.
<image1>
Which of the following statements about the angles is correct?

Choices:
- $\alpha=\beta$
- $2 \alpha+\beta=90$
- $\alpha+\beta=60$
- $2 \beta+\alpha=90$
- $\alpha+\beta=45$

Solution: The triangle $P Q R$ is congruent to the triangle $T Q S$ since they are right-angled triangles with sides of length 3 and 2 . Hence the angle $P Q R$ is also $\alpha^{\circ}$ and then $\alpha+\beta+\alpha=90$. One can check that the other statements are false.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-266.jpg?height=388&width=380&top_left_y=1565&top_left_x=1592)

metric geometry - angle

Question:
Two angles are marked on the $3 \times 3$ grid of squares.
<image1>
Which of the following statements about the angles is correct?
(A) $\alpha=\beta$
(B) $2 \alpha+\beta=90$
(C) $\alpha+\beta=60$
(D) $2 \beta+\alpha=90$
(E) $\alpha+\beta=45$


Question: All of the triangles in the figure and the central hexagon are equilateral. Given that $\overline{AC}$ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? <image1>

Answer: 3\sqrt{3}

Solution:
We divide the hexagon into six equilateral triangles, which are congruent by symmetry.  The star is made up of 12 of these triangles.    [asy]
pair A,B,C,D,E,F;
real x=sqrt(3);
F=(0,0);
E=(x,1);
D=(x,3);
C=(0,4);
A=(-x,1);
B=(-x,3);
draw(A--C--E--cycle); draw(B--D--F--cycle);
label("$D$",D,NE); label("$C$",C,N); label("$B$",B,NW); label("$A$",A,SW);
label("$F$",F,S); label("$E$",E,SE);
draw((1/x,1)--(-1/x,3)); draw((-1/x,1)--(1/x,3)); draw((2/x,2)--(-2/x,2));
[/asy] Let the side length of each triangle be $s$. $AC$ is made up of three triangle side lengths, so we have $3s=3 \Rightarrow s = 1$.  Thus, each triangle has area $\frac{1^2 \sqrt{3}}{4}$ and the star has area $12\cdot \frac{1^2 \sqrt{3}}{4} = \boxed{3\sqrt{3}}$.

metric geometry - area

Question: All of the triangles in the figure and the central hexagon are equilateral. Given that $\overline{AC}$ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? <image1>

Answer: 3\sqrt{3}


Question:
The faces of a die are either white, grey or black. Opposite faces are always a different colour. Which of the following nets does not belong to such a die?
<image1>

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

descriptive geometry

Question:
The faces of a die are either white, grey or black. Opposite faces are always a different colour. Which of the following nets does not belong to such a die?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
There are more grey squares than white. How many more?
<image1>

Answer: 9

counting

Question:
There are more grey squares than white. How many more?
<image1>

Answer: 9


Question: In isosceles triangle $ABC$, if $BC$ is extended to a point $X$ such that $AC = CX$, what is the number of degrees in the measure of angle $AXC$? <image1>

Answer: 15

Solution:
The angles opposite the equal sides of $\triangle ABC$ are congruent, so $\angle BCA=30^\circ$. Since $\angle BCA$ and $\angle XCA$ are supplementary, we have \begin{align*}
\angle XCA &= 180^\circ - \angle BCA\\
&= (180-30)^\circ \\
&= 150^\circ.
\end{align*} Since $\triangle ACX$ is isosceles with $AC=CX$, the angles $\angle XAC$ and $\angle AXC$ are congruent. Let each of them be $x^\circ$. Then the sum of angles in $\triangle ACX$ is $180^\circ$, so $$x + x + 150 = 180,$$ yielding $x=15$. That is, $\angle AXC = \boxed{15}$ degrees.

metric geometry - angle

Question: In isosceles triangle $ABC$, if $BC$ is extended to a point $X$ such that $AC = CX$, what is the number of degrees in the measure of angle $AXC$? <image1>

Answer: 15


Question:
A belt can be joined together in 5 different ways.
<image1>
How many $\mathrm{cm}$ is the belt longer if it is only closed in the first hole instead of in all 5 holes?
<image2>

Choices:
- $4 \mathrm{~cm}$
- $8 \mathrm{~cm}$
- $10 \mathrm{~cm}$
- $16 \mathrm{~cm}$
- $20 \mathrm{~cm}$

arithmetic

Question:
A belt can be joined together in 5 different ways.
<image1>
How many $\mathrm{cm}$ is the belt longer if it is only closed in the first hole instead of in all 5 holes?
<image2>
(A) $4 \mathrm{~cm}$
(B) $8 \mathrm{~cm}$
(C) $10 \mathrm{~cm}$
(D) $16 \mathrm{~cm}$
(E) $20 \mathrm{~cm}$


Question:
A kangaroo laid out 3 sticks like this to make a shape. It is not allowed to break or to bend the sticks. Which shape could the kangaroo make?
<image1>

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

combinatorial geometry

Question:
A kangaroo laid out 3 sticks like this to make a shape. It is not allowed to break or to bend the sticks. Which shape could the kangaroo make?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Mike sets the table for 8 people: The fork has to lie to the left and the knife to the right of the plate. For how many people is the cutlery set correctly?
<image1>

Answer: 5

counting

Question:
Mike sets the table for 8 people: The fork has to lie to the left and the knife to the right of the plate. For how many people is the cutlery set correctly?
<image1>

Answer: 5


Question:
Consider the five circles with midpoints $A, B, C, D$ and $E$ respectively, which touch each other as displayed in the diagram. The line segments, drawn in, connect the midpoints of adjacent circles. The distances between the midpoints are $A B=16, B C=14, C D=17, D E=13$ and $A E=14$ Which of the points is the midpoint of the circle with the biggest radius?
<image1>

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

metric geometry - length

Question:
Consider the five circles with midpoints $A, B, C, D$ and $E$ respectively, which touch each other as displayed in the diagram. The line segments, drawn in, connect the midpoints of adjacent circles. The distances between the midpoints are $A B=16, B C=14, C D=17, D E=13$ and $A E=14$ Which of the points is the midpoint of the circle with the biggest radius?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The diagram shows two identical large circles and two identical smaller circles whose centres are at the corners of a square. The two large circles are touching, and they each touch the two smaller circles. The radius of the small circles is $1 \mathrm{~cm}$. What is the radius of a large circle in centimetres? <image1>

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

Solution: Let $R$ be the radius of each of the larger circles. The sides of the square are equal to $R+1$, the sum of the two radii. The diagonal of the square is $2 R$. By Pythagoras, $(R+1)^{2}+(R+1)^{2}=(2 R)^{2}$. Simplifying gives $2(R+1)^{2}=4 R^{2}$, i.e. $(R+1)^{2}=2 R^{2}$, so $R+1=\sqrt{ } 2 R$ $[-\sqrt{2} R$ is not possible since $R+1>0]$. Therefore $(\sqrt{ } 2-1) R=1$. Hence $R=\frac{1}{\sqrt{2}-1}=\sqrt{2}+1$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-106.jpg?height=380&width=368&top_left_y=632&top_left_x=1461)

metric geometry - length

Question:
The diagram shows two identical large circles and two identical smaller circles whose centres are at the corners of a square. The two large circles are touching, and they each touch the two smaller circles. The radius of the small circles is $1 \mathrm{~cm}$. What is the radius of a large circle in centimetres? <image1>
(A) $1+\sqrt{2}$
(B) $\sqrt{5}$
(C) $\sqrt{2}$
(D) $\frac{5}{2}$
(E) $\frac{4}{5} \pi$


Question: The numbers on each pair of opposite faces on a die add up to 7 . A die is rolled without slipping around the circuit shown. At the start the top face is 3 . What number will be displayed on the top face at the end point? <image1>

Answer: 6

Solution: As the diagram in the question shows, after one rotation a 6 is on the top face After a second rotation 4 is on the top face. The next rotation brings the face that was on the back of the die to the top; since 2 was on the front this is 5 . A further rotation brings 3 to the top. With another rotation along the path, 1 is displayed, followed by 4 , and finally 6 .  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-046.jpg?height=383&width=423&top_left_y=1216&top_left_x=1359)

solid geometry

Question: The numbers on each pair of opposite faces on a die add up to 7 . A die is rolled without slipping around the circuit shown. At the start the top face is 3 . What number will be displayed on the top face at the end point? <image1>

Answer: 6


Question: Emily makes four identical numbered cubes using the net shown. She then glues them together so that only faces with the same number on are glued together to form the $2 \times 2 \times 1$ block shown. What is the largest possible total of all the numbers on the faces of the block that Emily could achieve? <image1>

Answer: 68

Solution: Each cube in the block has two adjacent faces that do not form part of the faces of the block. Since the numbers 1 and 2 are on opposite faces, it is not possible for both of these numbers to be hidden. However, the numbers 1 and 3 are on adjacent faces. Therefore, to obtain the largest possible total on the faces of the block, each cube will have numbers 1 and 3 hidden. Hence the largest possible total of the numbers on the faces on the block is $4 \times(6+5+4+2)=68$.

combinatorics

Question: Emily makes four identical numbered cubes using the net shown. She then glues them together so that only faces with the same number on are glued together to form the $2 \times 2 \times 1$ block shown. What is the largest possible total of all the numbers on the faces of the block that Emily could achieve? <image1>

Answer: 68


Question: The star on the right is formed from 12 identical equilateral triangles. The length of the perimeter of the star is $36 \mathrm{~cm}$. What is the length of the perimeter of the shaded hexagon? <image1>

Choices:
- $12 \mathrm{~cm}$
- $18 \mathrm{~cm}$
- $24 \mathrm{~cm}$
- $30 \mathrm{~cm}$
- $36 \mathrm{~cm}$

Solution: The perimeter of the star is formed from 12 sides of the equilateral triangles and that of the hexagon from 6 sides. So the perimeter of the hexagon is $\frac{1}{2} \times 36=18 \mathrm{~cm}$.

metric geometry - length

Question:
The star on the right is formed from 12 identical equilateral triangles. The length of the perimeter of the star is $36 \mathrm{~cm}$. What is the length of the perimeter of the shaded hexagon? <image1>
(A) $12 \mathrm{~cm}$
(B) $18 \mathrm{~cm}$
(C) $24 \mathrm{~cm}$
(D) $30 \mathrm{~cm}$
(E) $36 \mathrm{~cm}$


Question:
A child's toy hangs from the ceiling and it is in balance at all places. The same shapes have the same weight. The weight of a circle is 30 grams. What is the weight of a square?
<image1>

Answer: 20

algebra

Question:
A child's toy hangs from the ceiling and it is in balance at all places. The same shapes have the same weight. The weight of a circle is 30 grams. What is the weight of a square?
<image1>

Answer: 20


Question:
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

<image1>

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

metric geometry - area

Question:
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

<image1>
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $1$
(E) $\frac{\pi}{2}$


Question:
<image1>

The radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is

Choices:
- $\sqrt{2}$
- $\sqrt{1.25}$
- $1.25$
- $\frac{5\sqrt{17}}{16}$
- $\text{None of these}$

metric geometry - length

Question:
<image1>

The radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is
(A) $\sqrt{2}$
(B) $\sqrt{1.25}$
(C) $1.25$
(D) $\frac{5\sqrt{17}}{16}$
(E) $\text{None of these}$


Question:
Which of the following statements is definitely true for the angle marked in the diagram which is made up of nine squares?
<image1>

Choices:
- $\alpha=\beta$
- $2 \alpha+\beta=90^{\circ}$
- $\alpha+\beta=60^{\circ}$
- $2 \beta+\alpha=90^{\circ}$
- $\alpha+\beta=45^{\circ}$

metric geometry - angle

Question:
Which of the following statements is definitely true for the angle marked in the diagram which is made up of nine squares?
<image1>
(A) $\alpha=\beta$
(B) $2 \alpha+\beta=90^{\circ}$
(C) $\alpha+\beta=60^{\circ}$
(D) $2 \beta+\alpha=90^{\circ}$
(E) $\alpha+\beta=45^{\circ}$


Question: In the diagram, what is the area of $\triangle ABC$? <image1>

Answer: 54

Solution:
We think of $BC$ as the base of $\triangle ABC$. Its length is $12$.

Since the $y$-coordinate of $A$ is $9$, then the height of $\triangle ABC$ from base $BC$ is $9$.

Therefore, the area of $\triangle ABC$ is $\frac{1}{2} (12)(9) = \boxed{54}.$

analytic geometry

Question: In the diagram, what is the area of $\triangle ABC$? <image1>

Answer: 54


Question:
To meet the New Year day 2008, Basil put on a T-shirt with <image1> on it, and stood in front of a mirror on his hands, with his feet up. What number did Nick standing on his feet behind Basil see in the mirror?
<image2>

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

transformation geometry

Question:
To meet the New Year day 2008, Basil put on a T-shirt with <image1> on it, and stood in front of a mirror on his hands, with his feet up. What number did Nick standing on his feet behind Basil see in the mirror?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The large rectangle $W X Y Z$ is divided into seven identical rectangles, as shown. What is the ratio $W X: X Y$ ? <image1>

Choices:
- $3: 2$
- $4: 3$
- $8: 5$
- $12: 7$
- $7: 3$

Solution: Let the longer side of each of the small rectangles be $p$ and let the shorter side be $q$. From the diagram, it can be seen that $3 p=4 q$ and hence $q=\frac{3}{4} p$. It can also be seen that the ratio $W X: X Y=3 p: p+q$. This is equal to $3 p: p+\frac{3}{4} p=3 p: \frac{7}{4} p=12 p: 7 p=12: 7$.

metric geometry - length

Question:
The large rectangle $W X Y Z$ is divided into seven identical rectangles, as shown. What is the ratio $W X: X Y$ ? <image1>
(A) $3: 2$
(B) $4: 3$
(C) $8: 5$
(D) $12: 7$
(E) $7: 3$


Question:
A river starts at point $A$. As it flows the river splits into two. The first branch takes $\frac{2}{3}$ of the water and the second takes the rest. Later the first branch splits into three, one taking $\frac{1}{8}$ of the branch's water, the second $\frac{5}{8}$ and the third one the rest. Further down this last branch meets again a branch of the river. The map below shows the situation. What part of the original water flows at point $B$?
<image1>

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

graph theory

Question:
A river starts at point $A$. As it flows the river splits into two. The first branch takes $\frac{2}{3}$ of the water and the second takes the rest. Later the first branch splits into three, one taking $\frac{1}{8}$ of the branch's water, the second $\frac{5}{8}$ and the third one the rest. Further down this last branch meets again a branch of the river. The map below shows the situation. What part of the original water flows at point $B$?
<image1>
(A) $\frac{1}{3}$
(B) $\frac{5}{4}$
(C) $\frac{2}{9}$
(D) $\frac{1}{2}$
(E) $\frac{1}{4}$


Question:
Matias wrote 15 numbers on the wheel represented beside. Only one of them is visible, the 10 on top of the wheel. The sum of the numbers in any seven consecutive positions, such as the gray positions in the figure, does not vary. When seven numbers in consecutive positions are summed up, which of the following results is possible?
<image1>

Choices:
- 49
- 70
- 75
- 105
- 150

logic

Question:
Matias wrote 15 numbers on the wheel represented beside. Only one of them is visible, the 10 on top of the wheel. The sum of the numbers in any seven consecutive positions, such as the gray positions in the figure, does not vary. When seven numbers in consecutive positions are summed up, which of the following results is possible?
<image1>
(A) 49
(B) 70
(C) 75
(D) 105
(E) 150


Question:
On the number wall shown the number on each tile is equal to the sum of the numbers on the two tiles directly below it. Which number is on the tile marked with "?"?
<image1>

Answer: 16

algebra

Question:
On the number wall shown the number on each tile is equal to the sum of the numbers on the two tiles directly below it. Which number is on the tile marked with "?"?
<image1>

Answer: 16


Question:
Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters.  Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles.  The area of the square $ABCD$ is

<image1>

Answer: 64

metric geometry - area

Question:
Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters.  Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles.  The area of the square $ABCD$ is

<image1>

Answer: 64


Question:
An annulus is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a = XZ$, $ d = YZ$, and $ e = XY$. What is the area of the annulus?
<image1>

Choices:
- $\pi a^2$
- $\pi b^2$
- $\pi c^2$
- $\pi d^2$
- $\pi e^2$

metric geometry - area

Question:
An annulus is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a = XZ$, $ d = YZ$, and $ e = XY$. What is the area of the annulus?
<image1>
(A) $\pi a^2$
(B) $\pi b^2$
(C) $\pi c^2$
(D) $\pi d^2$
(E) $\pi e^2$


Question:
The $ 5\times 5$ grid shown contains a collection of squares with sizes from $ 1\times 1$ to $ 5\times 5$. How many of these squares contain the black center square?
<image1>

Answer: 19

combinatorics

Question:
The $ 5\times 5$ grid shown contains a collection of squares with sizes from $ 1\times 1$ to $ 5\times 5$. How many of these squares contain the black center square?
<image1>

Answer: 19


Question:
Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is

<image1>

Answer: 430

combinatorial geometry

Question:
Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is

<image1>

Answer: 430


Question:
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?

<image1>

Answer: 28

Solution:
Angle $\angle BCA$ and the angle we're trying to measure are alternate interior angles, so they are congruent. Thus, $\angle BCA=x^\circ$:

[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{\circ}$",(4.5,3),S);
label("$x^{\circ}$",(6,0),N);
[/asy]

Since $AB=BC$, we know that $\triangle ABC$ is isosceles with equal angles at $C$ and $A$. Therefore, $\angle BAC = x^\circ$:

[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{\circ}$",(4.5,3),S);
label("$x^{\circ}$",(6,0),N);
label("$x^{\circ}$",(3.6,1.7));
[/asy]

The sum of the three angles at $A$ is $180^\circ$, since they form a straight angle. Therefore, $$124+x+x=180,$$ which we can solve to obtain $x=\boxed{28}$.

metric geometry - angle

Question:
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?

<image1>

Answer: 28


Question:
$\textbf{Bake Sale}$
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.

$\circ$ Art's cookies are trapezoids:
<image1>

$\circ$ Roger's cookies are rectangles:
<image2>

$\circ$ Paul's cookies are parallelograms:
<image3>

$\circ$ Trisha's cookies are triangles:
<image4>

Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough?

Choices:
- $\text{Art}$
- $\text{Roger}$
- $\text{Paul}$
- $\text{Trisha}$
- $\text{There is a tie for fewest.}$

metric geometry - area

Question:
$\textbf{Bake Sale}$
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.

$\circ$ Art's cookies are trapezoids:
<image1>

$\circ$ Roger's cookies are rectangles:
<image2>

$\circ$ Paul's cookies are parallelograms:
<image3>

$\circ$ Trisha's cookies are triangles:
<image4>

Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough?
(A) $\text{Art}$
(B) $\text{Roger}$
(C) $\text{Paul}$
(D) $\text{Trisha}$
(E) $\text{There is a tie for fewest.}$


Question:
The suare in the diagram has side length 1. The radius of the small circle would
then be of the length
<image1>

Choices:
- $\sqrt{2}-1$
- $\frac{1}{4}$
- $\frac{\sqrt{2}}{4}$
- $1-\frac{\sqrt{2}}{2}$
- $(\sqrt{2}-1)^{2}$

metric geometry - length

Question:
The suare in the diagram has side length 1. The radius of the small circle would
then be of the length
<image1>
(A) $\sqrt{2}-1$
(B) $\frac{1}{4}$
(C) $\frac{\sqrt{2}}{4}$
(D) $1-\frac{\sqrt{2}}{2}$
(E) $(\sqrt{2}-1)^{2}$


Question:
In the diagram we see a cube and four marked angles. How big is the sum of those angles?
<image1>

Choices:
- $315^{\circ}$
- $330^{\circ}$
- $345^{\circ}$
- $360^{\circ}$
- $375^{\circ}$

solid geometry

Question:
In the diagram we see a cube and four marked angles. How big is the sum of those angles?
<image1>
(A) $315^{\circ}$
(B) $330^{\circ}$
(C) $345^{\circ}$
(D) $360^{\circ}$
(E) $375^{\circ}$


Question:
Which of the following diagrams is impossible to make with the two dominos?
<image1>
<image2>

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

combinatorial geometry

Question:
Which of the following diagrams is impossible to make with the two dominos?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the diagram a closed polygon can be seen whose vertices are the midpoints of the edges of the die. The interior angles are as usual defined as the angle that two sides of the polygon describe in a common vertex. How big is the sum of all interior angles of the polygon?
<image1>

Choices:
- $720^{\circ}$
- $1080^{\circ}$
- $1200^{\circ}$
- $1440^{\circ}$
- $1800^{\circ}$

solid geometry

Question:
In the diagram a closed polygon can be seen whose vertices are the midpoints of the edges of the die. The interior angles are as usual defined as the angle that two sides of the polygon describe in a common vertex. How big is the sum of all interior angles of the polygon?
<image1>
(A) $720^{\circ}$
(B) $1080^{\circ}$
(C) $1200^{\circ}$
(D) $1440^{\circ}$
(E) $1800^{\circ}$


Question:
In the figure below, $ABDC,$ $EFHG,$ and $ASHY$ are all squares; $AB=EF =1$ and $AY=5$.

What is the area of quadrilateral $DYES$?

<image1>

Answer: 15

Solution:
The large square, $ASHY$, is divided into seven regions. Two of these ($ABDC$ and $EFHG$) are squares. Four of the regions ($BSD,$ $CYD,$ $SFE,$ $YGE$) are right triangles. Finally, the seventh region is $DYES$, the quadrilateral whose area we wish to know. Thus, we subtract the area of the first six regions from the area of $ASHY$.

$ASHY$ has side length $5$ and therefore has area $5^2=25$.

The two small squares each have side length $1$ and thus area $1$.

The right triangles each have legs of length $1$ and $4$ (since, for example, $CY = AY-AC = 5-1 = 4$). Thus, each right triangle has area $\frac{1\cdot 4}{2} = 2$.

Subtracting the two small squares and the four right triangles from the large square, we determine that the area of $DYES$ is $$25 - 2\cdot(1) - 4\cdot(2) = \boxed{15}.$$

metric geometry - area

Question:
In the figure below, $ABDC,$ $EFHG,$ and $ASHY$ are all squares; $AB=EF =1$ and $AY=5$.

What is the area of quadrilateral $DYES$?

<image1>

Answer: 15


Question:
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
<image1>

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

analytic geometry

Question:
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
<image1>
(A) $\frac{1}{6}$
(B) $\frac{1}{5}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{1}{2}$


Question: The diagram shows 5 equally big semicircles and the length of 5 distances. How big is the radius of one semicircle? <image1>

Answer: 18

metric geometry - length

Question: The diagram shows 5 equally big semicircles and the length of 5 distances. How big is the radius of one semicircle? <image1>

Answer: 18