Question:
Ria wants to write a number in each of the seven bounded regions in the diagram. Two regions are neighbours if they share part of their boundary. The number in each region is to be the sum of the numbers in all of its neighbours. Ria has already written in two of the numbers, as shown.
What number must she write in the central region?
<image1>

Answer: 6

Solution: Let the numbers in the four regions that are neighbours to -4 be $a, b, c$ and $d$ as shown in the diagram. The question tells us that $a+b+c+d=-4$. However, we also know that $a+b+c+d+?=2$ and hence $?=6$.  (Note: The values $a=d=-4$ and $b=c=2$ give a  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-195.jpg?height=229&width=234&top_left_y=1730&top_left_x=1528) complete solution to the problem).

logic

Question:
Ria wants to write a number in each of the seven bounded regions in the diagram. Two regions are neighbours if they share part of their boundary. The number in each region is to be the sum of the numbers in all of its neighbours. Ria has already written in two of the numbers, as shown.
What number must she write in the central region?
<image1>

Answer: 6


Question:
A barcode as pictured is made up of alternate black and white stripes. The code always starts and ends with a black stripee. Each strip (black or white) has the width 1 or 2 and the total width of the barcode is 12. How many different barcodes if this kind are there if one reads from left to right?
<image1>

Answer: 116

combinatorics

Question:
A barcode as pictured is made up of alternate black and white stripes. The code always starts and ends with a black stripee. Each strip (black or white) has the width 1 or 2 and the total width of the barcode is 12. How many different barcodes if this kind are there if one reads from left to right?
<image1>

Answer: 116


Question:
I surrounded the wooden circle (see picture) using $a \mathrm{~cm}$ of thread. After that I surrounded by thread the wooden square –– $b \mathrm{~cm}$ of thread was enough for that. How much thread (in $\mathrm{cm}$ ) would be enough to surround the three wooden circles without moving them?
<image1>

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

metric geometry - length

Question:
I surrounded the wooden circle (see picture) using $a \mathrm{~cm}$ of thread. After that I surrounded by thread the wooden square –– $b \mathrm{~cm}$ of thread was enough for that. How much thread (in $\mathrm{cm}$ ) would be enough to surround the three wooden circles without moving them?
<image1>
(A) 3a
(B) 2a+b
(C) a+2b
(D) 3b
(E) a+b


Question:
Andrea has 4 equally long strips of paper. When she glues two together with an overlap of $10 \mathrm{~cm}$, she gets a strip $50 \mathrm{~cm}$ long.
<image1>
With the other two she wants to make a $56 \mathrm{~cm}$ long strip. How long must the overlap be?
<image2>

Choices:
- $4 \mathrm{~cm}$
- $6 \mathrm{~cm}$
- $8 \mathrm{~cm}$
- $10 \mathrm{~cm}$
- $12 \mathrm{~cm}$

arithmetic

Question:
Andrea has 4 equally long strips of paper. When she glues two together with an overlap of $10 \mathrm{~cm}$, she gets a strip $50 \mathrm{~cm}$ long.
<image1>
With the other two she wants to make a $56 \mathrm{~cm}$ long strip. How long must the overlap be?
<image2>
(A) $4 \mathrm{~cm}$
(B) $6 \mathrm{~cm}$
(C) $8 \mathrm{~cm}$
(D) $10 \mathrm{~cm}$
(E) $12 \mathrm{~cm}$


Question:
5 congruent rectangles are positioned in a square with side length 24 as shown in the diagram. How big is the area of one of these rectangles?
<image1>

Choices:
- $12 \mathrm{~cm}^{2}$
- $16 \mathrm{~cm}^{2}$
- $18 \mathrm{~cm}^{2}$
- $24 \mathrm{~cm}^{2}$
- $32 \mathrm{~cm}^{2}$

metric geometry - area

Question:
5 congruent rectangles are positioned in a square with side length 24 as shown in the diagram. How big is the area of one of these rectangles?
<image1>
(A) $12 \mathrm{~cm}^{2}$
(B) $16 \mathrm{~cm}^{2}$
(C) $18 \mathrm{~cm}^{2}$
(D) $24 \mathrm{~cm}^{2}$
(E) $32 \mathrm{~cm}^{2}$


Question:
The side of the square $A B C D$ is $10 \mathrm{~cm}$. The inner point $E$ of the square is such that $\angle E A B=75^{\circ}, \angle A B E=30^{\circ}$. The length of the segment $E C$ is:
<image1>

Choices:
- $8 \mathrm{~cm}$
- $9 \mathrm{~cm}$
- $9.5 \mathrm{~cm}$
- $10 \mathrm{~cm}$
- $11 \mathrm{~cm}$

metric geometry - length

Question:
The side of the square $A B C D$ is $10 \mathrm{~cm}$. The inner point $E$ of the square is such that $\angle E A B=75^{\circ}, \angle A B E=30^{\circ}$. The length of the segment $E C$ is:
<image1>
(A) $8 \mathrm{~cm}$
(B) $9 \mathrm{~cm}$
(C) $9.5 \mathrm{~cm}$
(D) $10 \mathrm{~cm}$
(E) $11 \mathrm{~cm}$


Question:
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides.  These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron.  What is the volume (in $\text{cm}^3$) of this polyhedron?

<image1>

Answer: 864

solid geometry

Question:
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides.  These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron.  What is the volume (in $\text{cm}^3$) of this polyhedron?

<image1>

Answer: 864


Question:
The line $P Q$ is divided into six parts by the points $V, W, X, Y$ and $Z$. Squares are drawn on $P V, V W, W X, X Y, Y Z$ and $Z Q$ as shown in the diagram. The length of line $P Q$ is $24 \mathrm{~cm}$. What is the length of the path from $P$ to $Q$ indicated by the arrows?
<image1>

Choices:
- $48 \mathrm{~cm}$
- $60 \mathrm{~cm}$
- $66 \mathrm{~cm}$
- $72 \mathrm{~cm}$
- $96 \mathrm{~cm}$

Solution: The path indicated follows three sides of each of the squares shown. The sum of the lengths of one side of each square is equal to the length of $P Q$, which is $24 \mathrm{~cm}$. Therefore the length of the path is $3 \times 24 \mathrm{~cm}=72 \mathrm{~cm}$.

metric geometry - length

Question:
The line $P Q$ is divided into six parts by the points $V, W, X, Y$ and $Z$. Squares are drawn on $P V, V W, W X, X Y, Y Z$ and $Z Q$ as shown in the diagram. The length of line $P Q$ is $24 \mathrm{~cm}$. What is the length of the path from $P$ to $Q$ indicated by the arrows?
<image1>
(A) $48 \mathrm{~cm}$
(B) $60 \mathrm{~cm}$
(C) $66 \mathrm{~cm}$
(D) $72 \mathrm{~cm}$
(E) $96 \mathrm{~cm}$


Question:
Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car N's speed and time are shown as solid line, which graph illustrates this?

<image1>

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

statistics

Question:
Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car N's speed and time are shown as solid line, which graph illustrates this?

<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the picture, three strips of the same horizontal width $a$ are marked 1,2,3. These strips connect the two parallel lines. Which strip has the biggest area?
<image1>

Choices:
- All three strips have the same area
- Strip 1
- Strip 2
- Strip 3
- Impossible to answer without knowing $a$

metric geometry - area

Question:
In the picture, three strips of the same horizontal width $a$ are marked 1,2,3. These strips connect the two parallel lines. Which strip has the biggest area?
<image1>
(A) All three strips have the same area
(B) Strip 1
(C) Strip 2
(D) Strip 3
(E) Impossible to answer without knowing $a$


Question:
The numbers in the five circles around each house add up to 20 . Some numbers are missing.
<image1>
Which number does the question mark stand for?

Answer: 9

algebra

Question:
The numbers in the five circles around each house add up to 20 . Some numbers are missing.
<image1>
Which number does the question mark stand for?

Answer: 9


Question:
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
<image1>

Answer: 31

algebra

Question:
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
<image1>

Answer: 31


Question:
<image1>

The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?

Answer: 189

combinatorial geometry

Question:
<image1>

The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?

Answer: 189


Question:
Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is

<image1>

Choices:
- $\sin \alpha$
- $\frac{1}{\sin \alpha}$
- $\frac{1}{1 - \cos \alpha}$
- $\frac{1}{\sin^2 \alpha}$
- $\frac{1}{(1 - \cos \alpha)^2}$

metric geometry - area

Question:
Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is

<image1>
(A) $\sin \alpha$
(B) $\frac{1}{\sin \alpha}$
(C) $\frac{1}{1 - \cos \alpha}$
(D) $\frac{1}{\sin^2 \alpha}$
(E) $\frac{1}{(1 - \cos \alpha)^2}$


Question:
Every time the witch has 3 apples she turns them in to 1 banana. Every time she has 3 bananas she turns them in to 1 apple. What will she finish with if she starts with 4 apples and 5 bananas?
<image1>
<image2>

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

algebra

Question:
Every time the witch has 3 apples she turns them in to 1 banana. Every time she has 3 bananas she turns them in to 1 apple. What will she finish with if she starts with 4 apples and 5 bananas?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The diagram shows a shape made from ten squares of side-length $1 \mathrm{~cm}$, joined edge to edge. What is the length of its perimeter, in centimetres?
<image1>

Answer: 18

Solution: Counting the edges of the squares around the shape gives a perimeter of $18 \mathrm{~cm}$.

combinatorial geometry

Question:
The diagram shows a shape made from ten squares of side-length $1 \mathrm{~cm}$, joined edge to edge. What is the length of its perimeter, in centimetres?
<image1>

Answer: 18


Question:
Maria drew the following figures on square sheets of paper.
<image1>
How many of these figures have the same perimeter as the square sheet of paper itself?

Answer: 4

metric geometry - length

Question:
Maria drew the following figures on square sheets of paper.
<image1>
How many of these figures have the same perimeter as the square sheet of paper itself?

Answer: 4


Question: Some mice live in three neighbouring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram? <image1>

Answer: 11

Solution: Let the number of mice that travelled from the left-hand house to the right-hand house along the bottom path be $x$ and the number of mice that travelled from the right-hand house to the left-hand house be $y$. Therefore, since every mouse changed house last night, $8-x$ mice travelled from the left-hand house to the top house and $7-y$ mice travelled from the right-hand house to the top house. Hence, since there were then 4 mice in the top house, we have $8-x+7-y=4$ and so $x+y=11$. Therefore the number of mice that travelled along the bottom path in the diagram is 11 .

counting

Question: Some mice live in three neighbouring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram? <image1>

Answer: 11


Question:
$A 10 \mathrm{~cm}$ long piece if wire is folded so that every part is equally long (see diagram). The wire is then cut through in the two positions marked. How long are the three pieces created in this way?
<image1>

Choices:
- $2 \mathrm{~cm}, 3 \mathrm{~cm}, 5 \mathrm{~cm}$
- $2 \mathrm{~cm}, 2 \mathrm{~cm}, 6 \mathrm{~cm}$
- $1 \mathrm{~cm}, 4 \mathrm{~cm}, 5 \mathrm{~cm}$
- $1 \mathrm{~cm}, 3 \mathrm{~cm}, 6 \mathrm{~cm}$
- $3 \mathrm{~cm}, 3 \mathrm{~cm}, 4 \mathrm{~cm}$

transformation geometry

Question:
$A 10 \mathrm{~cm}$ long piece if wire is folded so that every part is equally long (see diagram). The wire is then cut through in the two positions marked. How long are the three pieces created in this way?
<image1>
(A) $2 \mathrm{~cm}, 3 \mathrm{~cm}, 5 \mathrm{~cm}$
(B) $2 \mathrm{~cm}, 2 \mathrm{~cm}, 6 \mathrm{~cm}$
(C) $1 \mathrm{~cm}, 4 \mathrm{~cm}, 5 \mathrm{~cm}$
(D) $1 \mathrm{~cm}, 3 \mathrm{~cm}, 6 \mathrm{~cm}$
(E) $3 \mathrm{~cm}, 3 \mathrm{~cm}, 4 \mathrm{~cm}$


Question:
Lina has placed two tiles on a square game board. Which one of the 5 counters shown, can she add, so that none of the remaining four counters can be placed anymore?
<image1>
<image2>

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

descriptive geometry

Question:
Lina has placed two tiles on a square game board. Which one of the 5 counters shown, can she add, so that none of the remaining four counters can be placed anymore?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The cube pictured on the side is intersected by a plane that passes through the three points adjacent to $A$, that is $D, E$ and $B$. In a similar way the cube is also intersected by those planes that go through the three points adjacent to each of the other seven vertices. These planes dissect the cube into several pieces. What does the piece that contains the centre of the cube look like?
<image1>
<image2>

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

solid geometry

Question:
The cube pictured on the side is intersected by a plane that passes through the three points adjacent to $A$, that is $D, E$ and $B$. In a similar way the cube is also intersected by those planes that go through the three points adjacent to each of the other seven vertices. These planes dissect the cube into several pieces. What does the piece that contains the centre of the cube look like?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Four people can be seated at a square table. How many people at most could be seated if we pushed four tables of this kind together in one row?
<image1>

Answer: 10

arithmetic

Question:
Four people can be seated at a square table. How many people at most could be seated if we pushed four tables of this kind together in one row?
<image1>

Answer: 10


Question:
A $4 \times 1 \times 1$ cuboid is made up of 2 white and 2 grey cubes as shown. Which of the following cuboids can be build entirely out of such $4 \times 1 \times 1$ cuboids?
<image1>
<image2>

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

combinatorial geometry

Question:
A $4 \times 1 \times 1$ cuboid is made up of 2 white and 2 grey cubes as shown. Which of the following cuboids can be build entirely out of such $4 \times 1 \times 1$ cuboids?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A bike lock has four wheels numbered with the digits 0 to 9 in order. Each of the four wheels is rotated by $180^{\circ}$ from the code shown in the first diagram to get the correct code. What is the correct code for the bike lock?
<image1>
<image2>

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

transformation geometry

Question:
A bike lock has four wheels numbered with the digits 0 to 9 in order. Each of the four wheels is rotated by $180^{\circ}$ from the code shown in the first diagram to get the correct code. What is the correct code for the bike lock?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: In triangle $ABC, AB=AC, AE=AD$ and angle $BAD=30^{\circ}$. What is the size of angle $CDE$ ? <image1>

Choices:
- $10^{\circ}$
- $15^{\circ}$
- $20^{\circ}$
- $25^{\circ}$
- $30^{\circ}$

Solution: Let $x^{\circ}=\angle C D E, y^{\circ}=\angle E D A=\angle D E A, z^{\circ}=\angle D A E$.  The sum of the angles in triangle $A E D$ shows that $z=180-2 y$.  Then, from triangle $A B C, \angle A C B=\frac{1}{2}(180-(30+180-2 y))=y-15$. Now $\angle A E D=x+\angle A C B$, since it is exterior to triangle $D C E$. Hence $y=x+(y-15)$ and so $x=15$.  [Alternatively:  Let $x^{\circ}=\angle C D E, p^{\circ}=\angle A E D=\angle A D E$ and $q^{\circ}=\angle A B C=\angle A C B$.  Using the exterior angle theorem, for triangle $A B D$ we obtain $p+x=q+30$ and for triangle $C D E, p=q+x$.  Subtracting these equations gives $x=30-x$, so that $x=15$.]

metric geometry - angle

Question:
In triangle $ABC, AB=AC, AE=AD$ and angle $BAD=30^{\circ}$. What is the size of angle $CDE$ ? <image1>
(A) $10^{\circ}$
(B) $15^{\circ}$
(C) $20^{\circ}$
(D) $25^{\circ}$
(E) $30^{\circ}$


Question:
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
<image1>

Answer: 175

metric geometry - area

Question:
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
<image1>

Answer: 175


Question:
In the figure, formed by a square and an equilateral triangle, the letters indicate the measurements of the angles. Which of the following equality is true?
<image1>

Choices:
- $a=d$
- $b+c=d$
- $a+c=d+e$
- $a+b=d+e$
- $e+d=a$

metric geometry - angle

Question:
In the figure, formed by a square and an equilateral triangle, the letters indicate the measurements of the angles. Which of the following equality is true?
<image1>
(A) $a=d$
(B) $b+c=d$
(C) $a+c=d+e$
(D) $a+b=d+e$
(E) $e+d=a$


Question:
Jan sticks these three pieces of paper <image1> Which picture can he not obtain?
<image2>

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

combinatorial geometry

Question:
Jan sticks these three pieces of paper <image1> Which picture can he not obtain?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The regular eight-sided shape on the right has sides of length 10. A circle touches all inscribed diagonals of this eight-sided shape. What is the radius of this circle?
<image1>

Answer: 5

metric geometry - length

Question:
The regular eight-sided shape on the right has sides of length 10. A circle touches all inscribed diagonals of this eight-sided shape. What is the radius of this circle?
<image1>

Answer: 5


Question:
Carolina has a box of 30 matches. She begins to make the number 2022 using matchsticks. The diagram shows the first two digits.
<image1>
How many matchsticks will be left in the box when she has finished?

Answer: 9

Solution: Each of the twos in " 2022 " requires 5 matches, so 15 altogether. And she needs six matches for the zero, which leaves $30-15-6=9$ matches.

arithmetic

Question:
Carolina has a box of 30 matches. She begins to make the number 2022 using matchsticks. The diagram shows the first two digits.
<image1>
How many matchsticks will be left in the box when she has finished?

Answer: 9


Question: The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true? <image1>

Choices:
- $X, Y$ and $Z$ are all the same
- $\quad Y$ and $Z$ are equal, but $X$ is different
- $X$ and $Z$ are equal, but $Y$ is different
- $X$ and $Y$ are equal, but $Z$ is different
- $X, Y, Z$ are all different

Solution: By joining the vertices of the inner triangle to the centre of the hexagon in both the first and third diagrams, it can be seen that each hexagon has been dissected into six equal parts, three of which are shaded. Therefore, $X, Y$ and $Z$ are each half of the hexagon and hence they are all the same. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-251.jpg?height=212&width=750&top_left_y=1820&top_left_x=1096)

metric geometry - area

Question:
The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true? <image1>
(A) $X, Y$ and $Z$ are all the same
(B) $\quad Y$ and $Z$ are equal, but $X$ is different
(C) $X$ and $Z$ are equal, but $Y$ is different
(D) $X$ and $Y$ are equal, but $Z$ is different
(E) $X, Y, Z$ are all different


Question:
Andrea made the pattern in the picture out of several identical tiles. None of the tiles overlap each other. Which of the following tiles could she definitely not have used?
<image1>
<image2>

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

combinatorial geometry

Question:
Andrea made the pattern in the picture out of several identical tiles. None of the tiles overlap each other. Which of the following tiles could she definitely not have used?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Four circles of radius $3$ are arranged as shown.  Their centers are the vertices of a square.  The area of the shaded region is closest to

<image1>

Answer: 7.7

metric geometry - area

Question:
Four circles of radius $3$ are arranged as shown.  Their centers are the vertices of a square.  The area of the shaded region is closest to

<image1>

Answer: 7.7


Question:
In rectangle $ ABCD$, $ AD = 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$?
<image1>

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

metric geometry - length

Question:
In rectangle $ ABCD$, $ AD = 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$?
<image1>
(A) $3 + \frac{\sqrt{3}}{3}$
(B) $2 + \frac{4\sqrt{3}}{3}$
(C) $2 + 2\sqrt{2}$
(D) $\frac{3 + 3\sqrt{5}}{2}$
(E) $2 + \frac{5\sqrt{3}}{3}$


Question:
In a square of side $s$, where $s$ is an odd integer, the squares of side 1 on the diagonals are colored (like in the picture, where the square is of side 7). How many white squares are there?
<image1>

Choices:
- $s^{2}+1-2 s$
- $s^{2}+4-4 s$
- $2 s^{2}+1-4 s$
- $s^{2}-1-2 s$
- $s^{2}-2 s$

combinatorial geometry

Question:
In a square of side $s$, where $s$ is an odd integer, the squares of side 1 on the diagonals are colored (like in the picture, where the square is of side 7). How many white squares are there?
<image1>
(A) $s^{2}+1-2 s$
(B) $s^{2}+4-4 s$
(C) $2 s^{2}+1-4 s$
(D) $s^{2}-1-2 s$
(E) $s^{2}-2 s$


Question:
Jakob writes one of the natural numbers 1 to 9 into each cell of the $3 \times 3$-table. Then he works out the sum of the numbers in each row and in each column. Five of his results are 12, 13, 15, 16 and 17. What is the sixth sum?
<image1>

Answer: 17

algebra

Question:
Jakob writes one of the natural numbers 1 to 9 into each cell of the $3 \times 3$-table. Then he works out the sum of the numbers in each row and in each column. Five of his results are 12, 13, 15, 16 and 17. What is the sixth sum?
<image1>

Answer: 17


Question:
Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF = \sqrt{9 + 5\sqrt{2}}$. What is $ r/s$?
<image1>

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

metric geometry - length

Question:
Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF = \sqrt{9 + 5\sqrt{2}}$. What is $ r/s$?
<image1>
(A) $\frac{1}{2}$
(B) $\frac{5}{9}$
(C) $\frac{3}{5}$
(D) $\frac{5}{3}$
(E) $\frac{9}{5}$


Question:
The side lengths of the large regular hexagon are twice the length of those of the small regular hexagon. What is the area of the large hexagon if the small hexagon has an area of $4 \mathrm{~cm}^{2}$?
<image1>

Choices:
- $16 \mathrm{~cm}^{2}$
- $14 \mathrm{~cm}^{2}$
- $12 \mathrm{~cm}^{2}$
- $10 \mathrm{~cm}^{2}$
- $8 \mathrm{~cm}^{2}$

metric geometry - area

Question:
The side lengths of the large regular hexagon are twice the length of those of the small regular hexagon. What is the area of the large hexagon if the small hexagon has an area of $4 \mathrm{~cm}^{2}$?
<image1>
(A) $16 \mathrm{~cm}^{2}$
(B) $14 \mathrm{~cm}^{2}$
(C) $12 \mathrm{~cm}^{2}$
(D) $10 \mathrm{~cm}^{2}$
(E) $8 \mathrm{~cm}^{2}$


Question:
Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

<image1>

Answer: 4.0

metric geometry - area

Question:
Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

<image1>

Answer: 4.0


Question:
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number.  What is the smallest possible value of $s$?

<image1>

Answer: 4

metric geometry - length

Question:
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number.  What is the smallest possible value of $s$?

<image1>

Answer: 4


Question:
Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$.  Triangle $ABD$ has a right angle at $A$ and $AD = 12$.  Points $C$ and $D$ are on opposite sides of $\overline{AB}$.  The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$.  If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $
<image1>

Answer: 128

metric geometry - length

Question:
Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$.  Triangle $ABD$ has a right angle at $A$ and $AD = 12$.  Points $C$ and $D$ are on opposite sides of $\overline{AB}$.  The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$.  If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $
<image1>

Answer: 128


Question: Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)\n<image1>

Answer: 18

Solution: We claim that the answer is $\boxed{18}$. Observe that for any $2 \times 2$ grid, there can be a maximum of $2$ L-trominos with one square of each color. There are $9$ total $2 \times 2$ grids in the $4 \times 4$ grid, so the maximum possible answer is $2 \cdot 9 = 18$.\n\nFor a construction simply take ABAB / CCCC / ABAB / CCCC.

combinatorics

Question: Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)\n<image1>

Answer: 18


Question:
Marta sticks several triangles on top of each other and makes a star that way. What is the minimum number of triangles she has used?
<image1>

Answer: 3

combinatorial geometry

Question:
Marta sticks several triangles on top of each other and makes a star that way. What is the minimum number of triangles she has used?
<image1>

Answer: 3


Question:
Which of the following geometrical figures does not appear in the big picture?
<image1>
<image2>

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

combinatorial geometry

Question:
Which of the following geometrical figures does not appear in the big picture?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A large square is divided into smaller squares, as shown. A shaded circle is inscribed inside each of the smaller squares. What proportion of the area of the large square is shaded?
<image1>

Choices:
- $\frac{8 \pi}{9}$
- $\frac{13 \pi}{16}$
- $\frac{3}{\pi}$
- $\frac{3}{4}$
- $\frac{\pi}{4}$

metric geometry - area

Question:
A large square is divided into smaller squares, as shown. A shaded circle is inscribed inside each of the smaller squares. What proportion of the area of the large square is shaded?
<image1>
(A) $\frac{8 \pi}{9}$
(B) $\frac{13 \pi}{16}$
(C) $\frac{3}{\pi}$
(D) $\frac{3}{4}$
(E) $\frac{\pi}{4}$


Question:
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$.  One way to do this is shown below.  Find the number of positive integer divisors of $N$.

<image1>

Answer: 144

combinatorics

Question:
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$.  One way to do this is shown below.  Find the number of positive integer divisors of $N$.

<image1>

Answer: 144


Question:
<image1>
In the picture above five ladybirds can be seen. Each one is sitting on a certain flower. A ladybird is only allowed to sit on a flower if the following conditions are met:
1) The difference between the number of points on each wing is equal to the number of leaves on the stem.
2) The number of points on the wings of the ladybird is equal to the number of petals on the flower. Which of the following flowers is without a ladybird?
<image2>

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

logic

Question:
<image1>
In the picture above five ladybirds can be seen. Each one is sitting on a certain flower. A ladybird is only allowed to sit on a flower if the following conditions are met:
1) The difference between the number of points on each wing is equal to the number of leaves on the stem.
2) The number of points on the wings of the ladybird is equal to the number of petals on the flower. Which of the following flowers is without a ladybird?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Three triangles are connected to each other as shown. In which of the following pictures are the three triangles connected in the same way?
<image1>
<image2>

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

topology

Question:
Three triangles are connected to each other as shown. In which of the following pictures are the three triangles connected in the same way?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?

<image1>

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

metric geometry - length

Question:
Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?

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


Question:
In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB=52$, $ BC=12$, $ CD=39$, and $ DA=5$. The area of $ ABCD$ is

<image1>

Answer: 210

metric geometry - area

Question:
In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB=52$, $ BC=12$, $ CD=39$, and $ DA=5$. The area of $ ABCD$ is

<image1>

Answer: 210


Question:
The robot starts walking over white cells of the table from the cell A2 in the direction of the arrow, as shown in the picture. It goes always forward. If it meets an obstacle (a black cell or the border of the table), it turns right. The robot stops in case, it cannot go forward after turning right (i.e., it stops in the cell where the obstacles appear in front of him and on the right). In which cell will it stop?
<image1>

Choices:
- B2
- B1
- A1
- D1
- It never stops

graph theory

Question:
The robot starts walking over white cells of the table from the cell A2 in the direction of the arrow, as shown in the picture. It goes always forward. If it meets an obstacle (a black cell or the border of the table), it turns right. The robot stops in case, it cannot go forward after turning right (i.e., it stops in the cell where the obstacles appear in front of him and on the right). In which cell will it stop?
<image1>
(A) B2
(B) B1
(C) A1
(D) D1
(E) It never stops


Question:
In $\triangle{ABC}$, shown, $\cos{B}=\frac{3}{5}$.  What is $\cos{C}$?

<image1>

Answer: \frac{4}{5}

Solution: Since $\cos{B}=\frac{3}{5}$, we have $\cos{B}=\frac{AB}{BC}=\frac{9}{BC}=\frac{3}{5}$.  Then, we can see that we have $9=BC\cdot\frac{3}{5}$, so $BC=9\cdot\frac{5}{3}=15$.  From the Pythagorean Theorem, we have $AC=\sqrt{BC^2-AB^2}=\sqrt{15^2-9^2}=\sqrt{144}=12$.  Finally, we can find $\cos{C}$: $\cos{C}=\frac{AC}{BC}=\frac{12}{15}=\boxed{\frac{4}{5}}$.

metric geometry - angle

Question:
In $\triangle{ABC}$, shown, $\cos{B}=\frac{3}{5}$.  What is $\cos{C}$?

<image1>

Answer: \frac{4}{5}


Question:
This picture shows a bracelet with pearls.
<image1>
Which of the bands below shows the same bracelet as above?
<image2>

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

topology

Question:
This picture shows a bracelet with pearls.
<image1>
Which of the bands below shows the same bracelet as above?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Ellis's Eel Emporium contains a large tank holding three different types of eel: electric eels, moray eels and freshwater eels. A notice on the tank reads as follows:
<image1>
How many eels are in the tank?

Answer: 21

Solution: Let the number of electric eels be $x$, the number of moray eels be $y$ and the number of freshwater eels be $z$. The information on the notice tells us that $y+z=12, x+z=14$ and $x+y=16$. When you add these three equations, you obtain $2 x+2 y+2 z=42$ and hence $x+y+z=21$. Therefore the number of eels in the tank is 21 .

algebra

Question:
Ellis's Eel Emporium contains a large tank holding three different types of eel: electric eels, moray eels and freshwater eels. A notice on the tank reads as follows:
<image1>
How many eels are in the tank?

Answer: 21


Question:
Gerda walks along the road and writes down the letters she can see on her right hand side. Which word is formed while Gerda walks from point 1 to point 2?
<image1>

Choices:
- KNAO
- KNGO
- KNR
- AGRO
- KAO

combinatorics

Question:
Gerda walks along the road and writes down the letters she can see on her right hand side. Which word is formed while Gerda walks from point 1 to point 2?
<image1>
(A) KNAO
(B) KNGO
(C) KNR
(D) AGRO
(E) KAO


Question:
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\frac{a-\sqrt{2}}{b}$, where $a$ and $b$ are positive integers. What is $a+b$ ?

<image1>

Answer: 11

metric geometry - length

Question:
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\frac{a-\sqrt{2}}{b}$, where $a$ and $b$ are positive integers. What is $a+b$ ?

<image1>

Answer: 11


Question:
Florian has 10 equally long metal strips with equally many holes.
<image1>
He bolts the metal strips together in pairs. Now he has five long strips (see the diagram).
<image2>
Which of the long strips is the shortest?

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

counting

Question:
Florian has 10 equally long metal strips with equally many holes.
<image1>
He bolts the metal strips together in pairs. Now he has five long strips (see the diagram).
<image2>
Which of the long strips is the shortest?
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Identical regular pentagons are arranged in a ring. The partially completed ring is shown in the diagram. Each of the regular pentagons has a perimeter of 65 . The regular polygon formed as the inner boundary of the ring has a perimeter of $P$. What is the value of $P$ ?
<image1>

Answer: 130

Solution: Let the regular $N$-gon at the centre of the figure have interior angles of size $x$ degrees. The interior angle of a pentagon is $108^{\circ}$. By angles at a point we have $x+2 \times 108=360$, so $x=144$. The exterior angle of the $N$-gon is $180-144=36$. Therefore, the $N$-gon has $\frac{360}{36}=10$ sides. As each side has length $\frac{65}{5}=13$, the perimeter is $10 \times 13=130$.

metric geometry - length

Question:
Identical regular pentagons are arranged in a ring. The partially completed ring is shown in the diagram. Each of the regular pentagons has a perimeter of 65 . The regular polygon formed as the inner boundary of the ring has a perimeter of $P$. What is the value of $P$ ?
<image1>

Answer: 130


Question:
The numbers 1 to 7 should be written in the small circles so that the sum of the numbers along each line is the same. Which number should be written in the uppermost circle on the triangle?
<image1>

Answer: 4

logic

Question:
The numbers 1 to 7 should be written in the small circles so that the sum of the numbers along each line is the same. Which number should be written in the uppermost circle on the triangle?
<image1>

Answer: 4


Question: In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle's sides. If the area of the largest triangle is $64$, what is the area of the shaded region?\n<image1>

Answer: 15

Solution: First, see that the largest triangle's area is 64. The edge length of the triangle inscribed inside the largest triangle is half that of the largest triangle. Thus, the triangle inscribed inside the largest triangle is $\frac{1}{4}$ the area of the largest triangle because both its base and the height will be $\frac{1}{2}$ the base and height of the large triangle. This relationship holds true for the next inscribed triangle, and so on. Thus, the area of the smallest white triangle in the middle is $64 \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4}=1$ and the black region has area of $64 \cdot \frac{1}{4}-1=15$.

metric geometry - area

Question: In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle's sides. If the area of the largest triangle is $64$, what is the area of the shaded region?\n<image1>

Answer: 15


Question:
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle.  The circles are tangent to one another and to the sides of the rectangle as shown in the diagram.  The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers.  Find $p+q$.

<image1>

Answer: 154

metric geometry - length

Question:
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle.  The circles are tangent to one another and to the sides of the rectangle as shown in the diagram.  The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers.  Find $p+q$.

<image1>

Answer: 154


Question: A part of a polynomial of degree five is illegible due to an ink stain. It is known that all zeros of the polynomial are integers. What is the highest power of $x-1$ that divides this polynomial? <image1>

Choices:
- $(x-1)^{1}$
- $(x-1)^{2}$
- $(x-1)^{3}$
- $(x-1)^{4}$
- $(x-1)^{5}$

algebra

Question:
A part of a polynomial of degree five is illegible due to an ink stain. It is known that all zeros of the polynomial are integers. What is the highest power of $x-1$ that divides this polynomial? <image1>
(A) $(x-1)^{1}$
(B) $(x-1)^{2}$
(C) $(x-1)^{3}$
(D) $(x-1)^{4}$
(E) $(x-1)^{5}$


Question:
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.

<image1>

Choices:
- $2:3$
- $3:2$
- $6:\pi$
- $9: \pi$
- $30 : \pi$

metric geometry - area

Question:
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.

<image1>
(A) $2:3$
(B) $3:2$
(C) $6:\pi$
(D) $9: \pi$
(E) $30 : \pi$


Question:
In the picture the distance $KM=10, LN=15, KN=22$. Find the distance $LM$.
<image1>

Answer: 3

metric geometry - length

Question:
In the picture the distance $KM=10, LN=15, KN=22$. Find the distance $LM$.
<image1>

Answer: 3


Question:
In the figure, an arrow pointing from one person to another means that the first person is shorter than the second. For example, person $B$ is shorter than person $A$. Which person is the tallest?
<image1>

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

logic

Question:
In the figure, an arrow pointing from one person to another means that the first person is shorter than the second. For example, person $B$ is shorter than person $A$. Which person is the tallest?
<image1>
(A) Person A
(B) Person B
(C) Person C
(D) Person D
(E) Person E


Question:
A square is placed in a co-ordinate system as shown. Each point $(x \mid y)$ of the square is deleted and replaced by the point $\left(\frac{1}{x} \mid \frac{1}{y}\right)$. Which diagram shows the resulting shape?
<image1>
<image2>

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

analytic geometry

Question:
A square is placed in a co-ordinate system as shown. Each point $(x \mid y)$ of the square is deleted and replaced by the point $\left(\frac{1}{x} \mid \frac{1}{y}\right)$. Which diagram shows the resulting shape?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The diagram shows the net of an octahedron. Which edge meets the edge labelled with $\mathrm{x}$ if the net is folded up to form an octahedron?
<image1>

Answer: 5

solid geometry

Question:
The diagram shows the net of an octahedron. Which edge meets the edge labelled with $\mathrm{x}$ if the net is folded up to form an octahedron?
<image1>

Answer: 5


Question:
Susi folds a piece of paper in the middle. She stamps 2 holes.
<image1>
What does the piece of paper look like when she unfolds it again?
<image2>

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

transformation geometry

Question:
Susi folds a piece of paper in the middle. She stamps 2 holes.
<image1>
What does the piece of paper look like when she unfolds it again?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Charles cuts a rope into 3 equally long pieces. Then he makes one knot in one of the pieces, 2 in the next and in the third piece 3 knots. Then he lays the three pieces down in a random order. Which picture does he see?
<image1>

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

metric geometry - length

Question:
Charles cuts a rope into 3 equally long pieces. Then he makes one knot in one of the pieces, 2 in the next and in the third piece 3 knots. Then he lays the three pieces down in a random order. Which picture does he see?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Two semicircles are drawn as shown in the figure. The chord $C D$, of length 4 , is parallel to the diameter $A B$ of the greater semicircle and touches the smaller semicircle. Then the area of the shaded region equals
<image1>

Choices:
- $\pi$
- $1.5 \pi$
- $2 \pi$
- $3 \pi$
- Not enough data

metric geometry - area

Question:
Two semicircles are drawn as shown in the figure. The chord $C D$, of length 4 , is parallel to the diameter $A B$ of the greater semicircle and touches the smaller semicircle. Then the area of the shaded region equals
<image1>
(A) $\pi$
(B) $1.5 \pi$
(C) $2 \pi$
(D) $3 \pi$
(E) Not enough data


Question: The diagram shows a large equilateral triangle divided into 36 small equilateral triangles, each with area $1 \mathrm{~cm}^{2}$. What is the area of the shaded triangle, in $\mathrm{cm}^{2}$ ? <image1>

Answer: 11

Solution: For each of the parallelograms $A B C O, C D E O$, EFAO in the diagram, half of its area is from the shaded triangle. Hence the triangle is half of the hexagon formed by the three parallelograms. Since the hexagon is made of 22 triangles, the shaded triangle must have area $11 \mathrm{~cm}^{2}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-116.jpg?height=402&width=437&top_left_y=336&top_left_x=1392)

combinatorial geometry

Question: The diagram shows a large equilateral triangle divided into 36 small equilateral triangles, each with area $1 \mathrm{~cm}^{2}$. What is the area of the shaded triangle, in $\mathrm{cm}^{2}$ ? <image1>

Answer: 11


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:
The numbers 1, 2, 3 are written on the circumference of a circle. Then the sum of each pair of neighbouring numbers is written between them, so 6 numbers are obtained (1,3,2,5,3 and 4). This operation is repeated 4 more times, resulting in 96 numbers on the circle. What is the sum of these numbers?
<image1>

Answer: 486

algebra

Question:
The numbers 1, 2, 3 are written on the circumference of a circle. Then the sum of each pair of neighbouring numbers is written between them, so 6 numbers are obtained (1,3,2,5,3 and 4). This operation is repeated 4 more times, resulting in 96 numbers on the circle. What is the sum of these numbers?
<image1>

Answer: 486


Question:
A circular carpet is placed on a floor which is covered by equally big, square tiles. All tiles that have at least one point in common with the carpet are coloured in grey. Which of the following cannot be a result of this?
<image1>

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

combinatorial geometry

Question:
A circular carpet is placed on a floor which is covered by equally big, square tiles. All tiles that have at least one point in common with the carpet are coloured in grey. Which of the following cannot be a result of this?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?
<image1>

Choices:
- $\frac{3\sqrt{2}}{4}$
- $\frac{7\sqrt{6}}{16}$
- $\frac{\sqrt{5}}{2}$
- $\frac{2\sqrt{3}}{3}$
- $\frac{\sqrt{6}}{2}$

solid geometry

Question:
A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?
<image1>
(A) $\frac{3\sqrt{2}}{4}$
(B) $\frac{7\sqrt{6}}{16}$
(C) $\frac{\sqrt{5}}{2}$
(D) $\frac{2\sqrt{3}}{3}$
(E) $\frac{\sqrt{6}}{2}$


Question:
Which of the pictures below shows what you will see if you look from above the piece represented on the right?
<image1>
<image2>

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

descriptive geometry

Question:
Which of the pictures below shows what you will see if you look from above the piece represented on the right?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The following analog clock has two hands that can move independently of each other.
<image1>
Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.

Let $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.

Answer: 608

combinatorics

Question:
The following analog clock has two hands that can move independently of each other.
<image1>
Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.

Let $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.

Answer: 608


Question: The diagram shows three triangles which are formed by the five line segments $A C D F, B C G, G D E, A B$ and $E F$ so that $A C=B C=C D=G D=D F=E F$. Also $\angle C A B=\angle E F D$. What is the size, in degrees, of $\angle C A B$ ? <image1>

Answer: 60

Solution: Let the size in degrees of $\angle C A B$ be $x$. Since $A C=B C$, triangle $A B C$ is isosceles. Hence $\angle A B C=x^{\circ}$. Since angles in a triangle add to $180^{\circ}$, we have $\angle B C A=(180-2 x)^{\circ}$. Also, since vertically opposite angles are equal, we have $\angle G C D=(180-2 x)^{\circ}$.  The same argument can then be applied to triangle $C G D$ which is isosceles since $C D=G D$. Hence $\angle C D G=(180-2(180-2 x))^{\circ}=(4 x-180)^{\circ}$. Therefore, since vertically opposite angles are equal, $\angle F D E=(4 x-180)^{\circ}$.  The same argument can be applied once more, this time to triangle $F D E$, which is also isosceles since $D F=E F$. This gives $\angle E F D=(180-2(4 x-180))^{\circ}=(540-8 x)^{\circ}$.  However, we are also told that $\angle C A B=\angle E F D$ and hence $x=540-8 x$ or $9 x=540$. This has solution $x=60$ and hence the size, in degrees, of $\angle C A B$ is 60 .

metric geometry - angle

Question: The diagram shows three triangles which are formed by the five line segments $A C D F, B C G, G D E, A B$ and $E F$ so that $A C=B C=C D=G D=D F=E F$. Also $\angle C A B=\angle E F D$. What is the size, in degrees, of $\angle C A B$ ? <image1>

Answer: 60


Question: A circle inscribed in a square. Has two chords as shown in a pair. It has radius $2$, and $P$ bisects $TU$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle.\n<image1>

Answer: $2(\sqrt{2}-1)$

Solution: Let $O B$ intersect the circle at $X$ and $Y$, and the chord $P M$ at $Q$, such that $O$ lies between $X$ and $Q$. Then $M N X Q$ is a parallelogram. For, $O B \| N M$ by homothety at $C$ and $P M \| N X$ because $M N X P$ is an isoceles trapezoid. It follows that $Q X=M N$. Considering that the center of the circle together with points $M, C$, and $N$ determines a square of side length 2 , it follows that $M N=2 \sqrt{2}$, so the answer is $2 \sqrt{2}-2$.

metric geometry - length

Question: A circle inscribed in a square. Has two chords as shown in a pair. It has radius $2$, and $P$ bisects $TU$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle.\n<image1>

Answer: $2(\sqrt{2}-1)$


Question:
Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
<image1>

Choices:
- 8:17
- 25:49
- 36:25
- 32:17
- 36:17

combinatorial geometry

Question:
Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
<image1>
(A) 8:17
(B) 25:49
(C) 36:25
(D) 32:17
(E) 36:17


Question:
Each of the children Ali, Lea, Josef, Vittorio and Sophie get a birthday cake. The number on top of the cake shows how old the child is. Lea is two years older than Josef, but one year younger than Ali. Vittorio is the youngest. Which cake belongs to Sophie?
<image1>

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

logic

Question:
Each of the children Ali, Lea, Josef, Vittorio and Sophie get a birthday cake. The number on top of the cake shows how old the child is. Lea is two years older than Josef, but one year younger than Ali. Vittorio is the youngest. Which cake belongs to Sophie?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


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:
Six numbers are written on the following cards, as shown:
<image1>
What is the largest number you can form with the given cards?

Answer: 7685413092

algebra

Question:
Six numbers are written on the following cards, as shown:
<image1>
What is the largest number you can form with the given cards?

Answer: 7685413092


Question:
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?

<image1>

Answer: 4

descriptive geometry

Question:
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?

<image1>

Answer: 4


Question:
A white and a grey ring are interlinked with one another. Peter sees the two rings from the front as they are seen in the diagram on the right. Paul sees the rings from the back. What does he see?
<image1>
<image2>

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

descriptive geometry

Question:
A white and a grey ring are interlinked with one another. Peter sees the two rings from the front as they are seen in the diagram on the right. Paul sees the rings from the back. What does he see?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: In the diagram below, all seven of the small rectangles are congruent. If the perimeter of the large rectangle is $65$, what is its area?\n<image1>

Answer: 525/2

metric geometry - area

Question: In the diagram below, all seven of the small rectangles are congruent. If the perimeter of the large rectangle is $65$, what is its area?\n<image1>

Answer: 525/2


Question:
<image1>

In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the
measure of $\measuredangle BAO$ is

Choices:
- $10^\circ$
- $15^\circ$
- $20^\circ$
- $25^\circ$
- $30^\circ$

metric geometry - angle

Question:
<image1>

In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the
measure of $\measuredangle BAO$ is
(A) $10^\circ$
(B) $15^\circ$
(C) $20^\circ$
(D) $25^\circ$
(E) $30^\circ$


Question:
In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$.  The length of $PC$ is

<image1>

Answer: 9

metric geometry - length

Question:
In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$.  The length of $PC$ is

<image1>

Answer: 9


Question:
The big rectangle is made up of five small rectangles (see diagram). Lukas wants to colour in the small rectangles in red, blue and yellow. Two rectangles next to each other should be coloured in different colours.
<image1> How many ways are there for Lukas to do that?

Answer: 6

graph theory

Question:
The big rectangle is made up of five small rectangles (see diagram). Lukas wants to colour in the small rectangles in red, blue and yellow. Two rectangles next to each other should be coloured in different colours.
<image1> How many ways are there for Lukas to do that?

Answer: 6


Question: In the diagram, $PRT$ and $QRS$ are straight lines. What is the value of $x$? <image1>

Answer: 55

Solution:
Since $PQ=QR$, we have $\angle QPR=\angle QRP$.

Since $\angle PQR + \angle QPR + \angle QRP = 180^\circ$, we have $40^\circ + 2(\angle QRP) = 180^\circ$, so $2(\angle QRP) = 140^\circ$ or $\angle QRP = 70^\circ$.

Since $\angle PRQ$ and $\angle SRT$ are vertical angles, we have $\angle SRT = \angle PRQ = 70^\circ$.

Since $RS=RT$, we have $\angle RST = \angle RTS = x^\circ$.

From $\triangle RST$, we have  $\angle SRT + \angle RST + \angle RTS = 180^\circ$, so $70^\circ + 2x^\circ = 180^\circ$ or $2x = 110$ or $x=\boxed{55}$.

metric geometry - angle

Question: In the diagram, $PRT$ and $QRS$ are straight lines. What is the value of $x$? <image1>

Answer: 55


Question:
A paper in the shape of a regular hexagon, as the one shown, is folded in such a way that the three marked corners touch each other at the centre of the hexagon. What is the obtained figure?
<image1>

Choices:
- Six corner star
- Dodecagon
- Hexagon
- Square
- Triangle

descriptive geometry

Question:
A paper in the shape of a regular hexagon, as the one shown, is folded in such a way that the three marked corners touch each other at the centre of the hexagon. What is the obtained figure?
<image1>
(A) Six corner star
(B) Dodecagon
(C) Hexagon
(D) Square
(E) Triangle


Question: In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$. <image1> What is the area of the shaded region?

Answer: 900\pi

Solution:
We know that $OA$ and $OB$ are each radii of the semi-circle with center $O$. Thus, $OA=OB=OC+CB=32+36=68$. Therefore, $AC=AO+OC=68+32=100$.

The semi-circle with center $K$ has radius $AK=\frac{1}{2}(AC)=\frac{1}{2}(100)=50$.

The shaded area is equal to the area of the largest semi-circle with center $O$, minus the combined areas of the two smaller unshaded semi-circles with centers $K$ and $M$.  The radius of the smaller unshaded circle is $MB=\frac{1}{2}(CB)=\frac{1}{2}(36)=18$.  Therefore, the shaded area equals \begin{align*}
&\frac{1}{2}\pi(OB)^2-\left(\frac{1}{2}\pi(AK)^2+\frac{1}{2}\pi(MB)^2\right)\\
&=\frac{1}{2}\pi(68)^2-\left(\frac{1}{2}\pi(50)^2+\frac{1}{2}\pi(18)^2\right)\\
&=\frac{1}{2}\pi(68^2-50^2-18^2)\\
&=\frac{1}{2}\pi(4624-2500-324)\\
&=\frac{1}{2}\pi(1800)\\
&=\boxed{900\pi}
\end{align*}

metric geometry - area

Question: In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$. <image1> What is the area of the shaded region?

Answer: 900\pi


Question: In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?\n<image1>

Answer: 372

Solution: For each dot in the diagram, we can count the number of paths from January 1 to it by adding the number of ways to get to the dots to the left of it, above it, and above and to the left of it, starting from the topmost leftmost dot. This yields the following numbers of paths:\n\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\hline \multirow[t]{2}{*}{ Jan. 1 --> } & $* 1$ & $* 1$ & $* 1$ & $* 1$ & $* 1$ & $* 1$ & $* 1$ & $* 1$ & $* 1$ & $* 1$ \\n\hline & $* 1$ & $* 3$ & & $* 2$ & $* 4$ & & $* 2$ & $* 4$ & & $* 2$ \\n\hline & $* 1$ & $* 5$ & $* 8$ & $* 10$ & $* 16$ & $* 20$ & $* 22$ & $* 28$ & $* 32$ & $* 34$ \\n\hline & $* 1$ & & & *18 & & & $* 42$ & $*92$ & & $* 66$ \\n\hline & $*1$ & $* 2$ & $* 2$ & $*20$ & $*38$ & $*38$ & $* 80$ & $*214$ & $*306$ & $*372$&<-- Dec.31 \\n\hline\n\end{tabular}\nSo the number of paths from January 1 to December 31 is 372 .

combinatorics

Question: In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?\n<image1>

Answer: 372


Question:
Triangle RZT is generated by rotating the equilateral triangle AZC about point Z. Angle $\beta=\angle \mathrm{CZR}=70^{\circ}$. Determine angle $\alpha=\angle \mathrm{CAR}$.
<image1>

Choices:
- $20^{\circ}$
- $25^{\circ}$
- $30^{\circ}$
- $35^{\circ}$
- $40^{\circ}$

metric geometry - angle

Question:
Triangle RZT is generated by rotating the equilateral triangle AZC about point Z. Angle $\beta=\angle \mathrm{CZR}=70^{\circ}$. Determine angle $\alpha=\angle \mathrm{CAR}$.
<image1>
(A) $20^{\circ}$
(B) $25^{\circ}$
(C) $30^{\circ}$
(D) $35^{\circ}$
(E) $40^{\circ}$


Question:
In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$.  They intersect the hypotenuse at $ M$ and $ N$.  Then, $ MN$ has length:

<image1>

Choices:
- $2$
- $\frac{13}{5}$
- $3$
- $4$
- $\frac{24}{5}$

metric geometry - length

Question:
In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$.  They intersect the hypotenuse at $ M$ and $ N$.  Then, $ MN$ has length:

<image1>
(A) $2$
(B) $\frac{13}{5}$
(C) $3$
(D) $4$
(E) $\frac{24}{5}$


Question:
Numbers are to be placed into the square grid shown, so that each of the numbers $1,2,3,4$ and 5 appears exactly once in each row and in each column. Furthermore sthe sum of all numbers in the three black-bordered sections should always be the same. Which number has to be written into the top right cell?
<image1>

Answer: 3

logic

Question:
Numbers are to be placed into the square grid shown, so that each of the numbers $1,2,3,4$ and 5 appears exactly once in each row and in each column. Furthermore sthe sum of all numbers in the three black-bordered sections should always be the same. Which number has to be written into the top right cell?
<image1>

Answer: 3


Question:
The lateral surface area of the frustum of a solid right cone is the product of one-half the slant height ($L$) and the sum of the circumferences of the two circular faces. What is the number of square centimeters in the total surface area of the frustum shown here? Express your answer in terms of $\pi$.

<image1>

Answer: 256\pi

Solution:
The circumferences of the bases are $2 \pi \cdot 4 = 8 \pi$ and $2 \pi \cdot 10 = 20 \pi$.  To find the slant height, we drop perpendiculars.

[asy]
unitsize(0.3 cm);

draw((-10,0)--(10,0)--(4,8)--(-4,8)--cycle);
draw((4,0)--(4,8));
draw((-4,0)--(-4,8));

label("$8$", (0,0), S);
label("$6$", (7,0), S);
label("$6$", (-7,0), S);
label("$8$", (0,8), N);
label("$8$", (4,4), W);
label("$L$", (7,4), NE);
[/asy]

We have created a right triangle with legs 6 and 8, so the hypotenuse is $L = 10$.

Hence, the total surface area of the frustum, including the two bases, is \[\pi \cdot 4^2 + \pi \cdot 10^2 + \frac{1}{2} \cdot 10 \cdot (8 \pi + 20 \pi) = \boxed{256 \pi}.\]

solid geometry

Question:
The lateral surface area of the frustum of a solid right cone is the product of one-half the slant height ($L$) and the sum of the circumferences of the two circular faces. What is the number of square centimeters in the total surface area of the frustum shown here? Express your answer in terms of $\pi$.

<image1>

Answer: 256\pi


Question:
Gaspar has these seven different pieces, formed by equal little squares.
<image1>
He uses all these pieces to assemble rectangles with different perimeters, that is, with different shapes. How many different perimeters can he find?

Answer: 3

combinatorial geometry

Question:
Gaspar has these seven different pieces, formed by equal little squares.
<image1>
He uses all these pieces to assemble rectangles with different perimeters, that is, with different shapes. How many different perimeters can he find?

Answer: 3


Answer: 15

arithmetic

Answer: 15


Question:
Mike has 125 small, equally big cubes. He glues some of them together in such a way that one big cube with exactly nine tunnels is created (see diagram). The tunnels go all the way straight through the cube. How many of the 125 cubes is he not using?
<image1>

Answer: 39

solid geometry

Question:
Mike has 125 small, equally big cubes. He glues some of them together in such a way that one big cube with exactly nine tunnels is created (see diagram). The tunnels go all the way straight through the cube. How many of the 125 cubes is he not using?
<image1>

Answer: 39


Question:
An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
<image1>

Choices:
- $80-20\pi$
- $60-10\pi$
- $80-10\pi$
- $60+10\pi$
- $80+10\pi$

metric geometry - length

Question:
An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
<image1>
(A) $80-20\pi$
(B) $60-10\pi$
(C) $80-10\pi$
(D) $60+10\pi$
(E) $80+10\pi$


Question: Two right triangles share a side as follows: <image1> What is the area of $\triangle ABE$?

Answer: \frac{40}{9}

Solution:
Since $AB = BD,$ we see that $\triangle ABD$ is an isosceles right triangle, therefore $\angle DAB = 45^\circ.$ That means that $AD$, and consequently $AE,$ bisects $\angle CAB.$

Relating our areas to side lengths and applying the Angle Bisector Theorem, we have that: \begin{align*}
\frac{[\triangle AEC]}{[\triangle ABE]} &= \frac{CE}{EB} = \frac{CA}{AB} \\
\frac{[\triangle AEC]}{[\triangle ABE]} + 1 &= \frac{CA}{AB} + 1 \\
\frac{[\triangle AEC] + [\triangle ABE]}{[\triangle ABE]} &= \frac{CA + AB}{AB} \\
\frac{[\triangle ABC]}{[\triangle ABE]} &= \frac{5 + 4}{4} = \frac{9}{4}.
\end{align*} Now, we see that $[\triangle ABC] = \frac{1}{2} \cdot 4 \cdot 5 = 10,$ so $[\triangle ABE] = \frac{4}{9} \cdot [\triangle ABC] = \frac{4}{9} \cdot 10 = \boxed{\frac{40}{9}}.$

metric geometry - area

Question: Two right triangles share a side as follows: <image1> What is the area of $\triangle ABE$?

Answer: \frac{40}{9}


Question:
In a game of luck, A ball rolls downwards towards hammered nails and is diverted either to the right or the left by a nail immediately below it. One possible path is shown in the diagram. How many different ways are there for the ball to reach the second compartment from the left?
<image1>

Answer: 4

combinatorics

Question:
In a game of luck, A ball rolls downwards towards hammered nails and is diverted either to the right or the left by a nail immediately below it. One possible path is shown in the diagram. How many different ways are there for the ball to reach the second compartment from the left?
<image1>

Answer: 4


Question:
A big $ L$ is formed as shown. What is its area?
<image1>

Answer: 22

metric geometry - area

Question:
A big $ L$ is formed as shown. What is its area?
<image1>

Answer: 22


Question:
The two circles shown on the right intersect each other at $X$ and $Y$. Thereby $X Y$ is the diameter of the small circle. The centre $S$ of the large circle (with radius $r$ ) is on the small circle. How big is the area of the grey region?
<image1>

Choices:
- $\frac{\pi}{6} r^{2}$
- $\frac{\sqrt{3} \pi}{12} r^{2}$
- $\frac{1}{2} r^{2}$
- $\frac{\sqrt{3}}{4} r^{2}$
- another number

metric geometry - area

Question:
The two circles shown on the right intersect each other at $X$ and $Y$. Thereby $X Y$ is the diameter of the small circle. The centre $S$ of the large circle (with radius $r$ ) is on the small circle. How big is the area of the grey region?
<image1>
(A) $\frac{\pi}{6} r^{2}$
(B) $\frac{\sqrt{3} \pi}{12} r^{2}$
(C) $\frac{1}{2} r^{2}$
(D) $\frac{\sqrt{3}}{4} r^{2}$
(E) another number


Question:
The section of a cube by a plane generates a plane figure. I have plotted this section in the development of the cube (see the picture). Can you find out what figure it is?
<image1>

Choices:
- An equilateral triangle
- A rectangle, but not a square
- A right triangle
- A square
- A hexagon

descriptive geometry

Question:
The section of a cube by a plane generates a plane figure. I have plotted this section in the development of the cube (see the picture). Can you find out what figure it is?
<image1>
(A) An equilateral triangle
(B) A rectangle, but not a square
(C) A right triangle
(D) A square
(E) A hexagon


Question:
The rear window wiper of a car is made in a way so that the rod $r$ and the wiper blade $\mathrm{w}$ are equally long and are connected at an angle $\alpha$. The wiper rotates around the centre of rotation $\mathrm{O}$ and wipes over the area shown on the right. Calculate the angle $\beta$ between the right edge of the cleaned area and the tangent of the curved upper edge.
<image1>

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

metric geometry - angle

Question:
The rear window wiper of a car is made in a way so that the rod $r$ and the wiper blade $\mathrm{w}$ are equally long and are connected at an angle $\alpha$. The wiper rotates around the centre of rotation $\mathrm{O}$ and wipes over the area shown on the right. Calculate the angle $\beta$ between the right edge of the cleaned area and the tangent of the curved upper edge.
<image1>
(A) $\frac{3 \pi-\alpha}{2}$
(B) $\pi-\frac{\alpha}{2}$
(C) $\frac{3 \pi}{2}-\alpha$
(D) $\frac{\pi}{2}+\alpha$
(E) $\pi+\frac{\alpha}{2}$


Question: Two circles are centered at the origin, as shown.  The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle.  If $QR=3$, what is the value of $k$? <image1>

Answer: 7

Solution:
We can determine the distance from $O$ to $P$ by dropping a perpendicular from $P$ to $T$ on the $x$-axis. [asy]
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;
draw(Circle((0,0),7)); draw(Circle((0,0),10));
dot((0,0)); dot((7,0)); dot((10,0)); dot((0,7)); dot((8,6));
draw((0,0)--(8,6)--(8,0));
label("$S (0,k)$",(0,7.5),W);
draw((13,0)--(0,0)--(0,13),Arrows(TeXHead));
draw((-13,0)--(0,0)--(0,-13));
draw((8.8,0)--(8.8,.8)--(8,.8));
label("$x$",(13,0),E); label("$y$",(0,13),N); label("$P(8,6)$",(8,6),NE);

label("$O$",(0,0),SW); label("$Q$",(7,0),SW); label("$T$",(8,0),S); label("$R$",(10,0),SE);

[/asy] We have $OT=8$ and $PT=6$, so by the Pythagorean Theorem, \[ OP^2 = OT^2 + PT^2 = 8^2+6^2=64+36=100 \]Since $OP>0$, then $OP = \sqrt{100}=10$. Therefore, the radius of the larger circle is $10$. Thus, $OR=10$.

Since $QR=3$, then $OQ = OR - QR = 10 - 3 = 7$. Therefore, the radius of the smaller circle is $7$.

Since $S$ is on the positive $y$-axis and is 7 units from the origin, then the coordinates of $S$ are $(0,7)$, which means that $k=\boxed{7}$.

analytic geometry

Question: Two circles are centered at the origin, as shown.  The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle.  If $QR=3$, what is the value of $k$? <image1>

Answer: 7


Question:
In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD = BC = 5, AB = 4,$ and $DC = 10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in the right triangle $DEF$. Then $CF =$
<image1>

Answer: 4.0

metric geometry - length

Question:
In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD = BC = 5, AB = 4,$ and $DC = 10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in the right triangle $DEF$. Then $CF =$
<image1>

Answer: 4.0


Question: Emma should colour in the three strips of the flag shown. She has four colours available. She can only use one colour for each strip and immediately adjacent strips are not to be of the same colour. How many different ways are there for her to colour in the flag? <image1>

Answer: 36

graph theory

Question: Emma should colour in the three strips of the flag shown. She has four colours available. She can only use one colour for each strip and immediately adjacent strips are not to be of the same colour. How many different ways are there for her to colour in the flag? <image1>

Answer: 36


Question:
Each of the 4 vertices and 6 edges of a tetrahedron is labelled with one of the numbers $1,2,3,4,5,6,7,8,9$ and 11. (The number 10 is left out). Each number is only used once. The number on each edge is the sum of the numbers on the two vertices which are connected by that edge. The edge $A B$ has the number 9. With which number is the edge $C D$ labelled?
<image1>

Answer: 5

solid geometry

Question:
Each of the 4 vertices and 6 edges of a tetrahedron is labelled with one of the numbers $1,2,3,4,5,6,7,8,9$ and 11. (The number 10 is left out). Each number is only used once. The number on each edge is the sum of the numbers on the two vertices which are connected by that edge. The edge $A B$ has the number 9. With which number is the edge $C D$ labelled?
<image1>

Answer: 5


Question:
How many of the figures shown can be drawn with one continuous line without drawing a segment twice?
<image1>

Answer: 3

Solution: All are possible except for the second figure. Each of the small lines must either start or end the continuous line. But then only one of the semicircles in the second figure can be included.

graph theory

Question:
How many of the figures shown can be drawn with one continuous line without drawing a segment twice?
<image1>

Answer: 3


Question:
In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. <image1>

What is the degree measure of the smallest angle in triangle $PQS$?

Answer: 30

Solution:
Join $PQ$, $PR$, $PS$, $RQ$, and $RS$. Since the circles with center $Q$, $R$ and $S$ are all tangent to $BC$, then $QR$ and $RS$ are each parallel to $BC$ (as the centres $Q$, $R$ and $S$ are each 1 unit above $BC$). This tells us that $QS$ passes through $R$. When the centers of tangent circles are joined, the line segments formed pass through the associated point of tangency, and so have lengths equal to the sum of the radii of those circles. Therefore, $QR=RS=PR=PS=1+1=2$.

[asy]
size(200);
pair P, Q, R, S;
Q=(0,0);
R=(2,0);
S=(4,0);
P=(3,1.732);
label("Q", Q, SW);
label("R", R, dir(270));
label("S", S, SE);
label("P", P, N);
draw(circle(Q,1), dashed);
draw(circle(P,1), dashed);
draw(circle(R,1), dashed);
draw(circle(S,1), dashed);
draw(P--Q--S--P--R);
[/asy]

Since $PR=PS=RS$, we know $\triangle PRS$ is equilateral, so $\angle PSR=\angle PRS=60^\circ$.  Since $\angle PRS=60^\circ$ and $QRS$ is a straight line, we have $\angle QRP=180^\circ-60^\circ=120^\circ$. Since $QR=RP$, we know $\triangle QRP$ is isosceles, so $$\angle PQR = \frac{1}{2}(180^\circ-120^\circ)= 30^\circ.$$Since $\angle PQS=30^\circ$ and $\angle PSQ=60^\circ$, we have $\angle QPS = 180^\circ - 30^\circ - 60^\circ = 90^\circ$, so $\triangle PQS$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle. Thus, the answer is $\boxed{30^\circ}$.

metric geometry - angle

Question:
In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. <image1>

What is the degree measure of the smallest angle in triangle $PQS$?

Answer: 30


Question:
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$.  Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$.  Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$.  It follows that the sine of the minor arc $AB$ is a rational number.  If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$?
<image1>

Answer: 175

metric geometry - length

Question:
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$.  Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$.  Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$.  It follows that the sine of the minor arc $AB$ is a rational number.  If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$?
<image1>

Answer: 175


Question:
The next window is a square of area $1 \mathrm{~m}^{2}$ and is composed of four triangles, which areas, indicated in the figure, follow the ratios $3 A=4 B$ and $2 C=3 D$. A fly is placed exactly at the point where these four triangles touch each other. The fly flies directly to the side closest to the window. How much does it fly?
<image1>

Choices:
- $40 \mathrm{~cm}$
- $30 \mathrm{~cm}$
- $25 \mathrm{~cm}$
- $20 \mathrm{~cm}$
- $10 \mathrm{~cm}$

metric geometry - length

Question:
The next window is a square of area $1 \mathrm{~m}^{2}$ and is composed of four triangles, which areas, indicated in the figure, follow the ratios $3 A=4 B$ and $2 C=3 D$. A fly is placed exactly at the point where these four triangles touch each other. The fly flies directly to the side closest to the window. How much does it fly?
<image1>
(A) $40 \mathrm{~cm}$
(B) $30 \mathrm{~cm}$
(C) $25 \mathrm{~cm}$
(D) $20 \mathrm{~cm}$
(E) $10 \mathrm{~cm}$


Question:
Johannes wrote the numbers 6,7 and 8 in the circles as shown. He wants to write the numbers 1, 2, 3, 4 and 5 in the remaining circles so that the sum of the numbers along each side of the square is 13. What will be the sum of the numbers in the grey circles?
<image1>

Answer: 16

algebra

Question:
Johannes wrote the numbers 6,7 and 8 in the circles as shown. He wants to write the numbers 1, 2, 3, 4 and 5 in the remaining circles so that the sum of the numbers along each side of the square is 13. What will be the sum of the numbers in the grey circles?
<image1>

Answer: 16


Question:
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?

<image1>

Answer: 6

solid geometry

Question:
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?

<image1>

Answer: 6


Question:
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
<image1>

For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34  \to 17 \to 52 \to 26$$When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_  \to \_\_ \to \_\_  \to 1$$

Choices:
- 73
- 74
- 75
- 82
- 83

arithmetic

Question:
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
<image1>

For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34  \to 17 \to 52 \to 26$$When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_  \to \_\_ \to \_\_  \to 1$$
(A) 73
(B) 74
(C) 75
(D) 82
(E) 83


Question:
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes?
<image1>

Choices:
- (35,44)
- (36,45)
- (37,45)
- (44,35)
- (45,36)

algebra

Question:
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes?
<image1>
(A) (35,44)
(B) (36,45)
(C) (37,45)
(D) (44,35)
(E) (45,36)


Question: In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively.  Determine the area of quadrilateral $DBEF$. <image1>

Answer: 8

Solution:
$\triangle DBC$ has base $\overline{BC}$ of length 8 and height $\overline{BD}$ of length 3; therefore, its area is $\frac{1}{2}\times8\times 3=12$.

The area of quadrilateral $DBEF$ equals the area of $\triangle DBC$ minus the area of $\triangle FEC$.

$\triangle FEC$ has base $EC=BC-BE=8-4=4$. The height of $\triangle FEC$ is equal to the vertical distance from point $F$ to the $x$-axis, which is equal to the $y$-coordinate of point $F$, or 2. Therefore, the area of $\triangle FEC$ is $\frac{1}{2}\times4\times 2=4$.

Finally, the area of quadrilateral $DBEF$ is $12-4=\boxed{8}$.

analytic geometry

Question: In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively.  Determine the area of quadrilateral $DBEF$. <image1>

Answer: 8


Question:
Rosana has some balls of 3 different colours. Balls of the same colour have the same weight. What is the weight of each white ball $\bigcirc$ ?
<image1>

Answer: 5

algebra

Question:
Rosana has some balls of 3 different colours. Balls of the same colour have the same weight. What is the weight of each white ball $\bigcirc$ ?
<image1>

Answer: 5


Question:
The diagram on the right shows a large triangle divided up into squares and triangles. $S$ is the number of squares of any size in the diagram and $T$ is the number of triangles of any size in the diagram. What is the value of $S \times T$ ?
<image1>

Answer: 70

Solution: In the diagram, there are six $1 \times 1$ squares and one $2 \times 2$ square. There are also four triangles that are half of a $1 \times 1$ square, three triangles that are half of a $2 \times 2$ square, two triangles that are half of a $3 \times 3$ square and one triangle that is half of a $4 \times 4$ square. Hence $S=7$ and $T=10$ so $S \times T=70$.

combinatorics

Question:
The diagram on the right shows a large triangle divided up into squares and triangles. $S$ is the number of squares of any size in the diagram and $T$ is the number of triangles of any size in the diagram. What is the value of $S \times T$ ?
<image1>

Answer: 70


Question: The rectangle $A B C D$ lies below the $x$-axis, and to the left of the $y$-axis. The edges of the rectangle are parallel to the coordinate axes. For each point $A, B, C, D$, the $y$-coordinate is divided by the $x$-coordinate. Which of the points yields the smallest value from this calculation? <image1>

Choices:
- A
- C
- C
- D
- it depends on the size of the rectangle

Solution: Since all the coordinates are negative, each of the calculations will yield a positive value. The smallest value will come from the least negative $y$-coordinate divided by the most negative $x$-coordinate; this comes from point $A$.

analytic geometry

Question:
The rectangle $A B C D$ lies below the $x$-axis, and to the left of the $y$-axis. The edges of the rectangle are parallel to the coordinate axes. For each point $A, B, C, D$, the $y$-coordinate is divided by the $x$-coordinate. Which of the points yields the smallest value from this calculation? <image1>
(A) A
(B) C
(C) C
(D) D
(E) it depends on the size of the rectangle


Question:
Five big and four small elephants are marching along a path. Since the path is narrow the elephants cannot change their order. At the fork in the path each elephant either goes to the right or to the left. Which of the following situations cannot happen?
<image1>
<image2>

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

combinatorics

Question:
Five big and four small elephants are marching along a path. Since the path is narrow the elephants cannot change their order. At the fork in the path each elephant either goes to the right or to the left. Which of the following situations cannot happen?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A (very small) ball is kicked off from point A on a square billiard table with side length $2 \mathrm{~m}$. After moving along the shown path and touching the sides three times as indicated, the path ends in point $B$. How long is the path that the bal travels from A to B? (As indicated on the right: incident angle = emergent angle.)
<image1>

Choices:
- 7
- $2 \sqrt{13}$
- 8
- $4 \sqrt{3}$
- $2 \cdot(\sqrt{2}+\sqrt{3})$

metric geometry - length

Question:
A (very small) ball is kicked off from point A on a square billiard table with side length $2 \mathrm{~m}$. After moving along the shown path and touching the sides three times as indicated, the path ends in point $B$. How long is the path that the bal travels from A to B? (As indicated on the right: incident angle = emergent angle.)
<image1>
(A) 7
(B) $2 \sqrt{13}$
(C) 8
(D) $4 \sqrt{3}$
(E) $2 \cdot(\sqrt{2}+\sqrt{3})$


Question:
In the diagram to the right a $2 \times 2 \times 2$ cube is made up of four transparent $1 \times 1 \times 1$ cubes and four non-transparent black $1 \times 1 \times 1$ cubes. They are placed in a way so that the entire big cube is nontransparent; i.e. looking at it from the front to the back, the right to the left, the top to the bottom, at no point you can look through the cube. What is the minimum number of black $1 \times 1 \times 1$ cubes needed to make a $3 \times 3 \times 3$ cube non-transparent in the same way?
<image1>

Answer: 9

solid geometry

Question:
In the diagram to the right a $2 \times 2 \times 2$ cube is made up of four transparent $1 \times 1 \times 1$ cubes and four non-transparent black $1 \times 1 \times 1$ cubes. They are placed in a way so that the entire big cube is nontransparent; i.e. looking at it from the front to the back, the right to the left, the top to the bottom, at no point you can look through the cube. What is the minimum number of black $1 \times 1 \times 1$ cubes needed to make a $3 \times 3 \times 3$ cube non-transparent in the same way?
<image1>

Answer: 9


Question: What is the number of square centimeters in the shaded area? (The 10 represents the hypotenuse of the white triangle only.) <image1>

Answer: 30

Solution:
We start by labeling everything first: [asy]
fill((6,0)--(9,0)--(9,12)--(6,8)--cycle,gray(0.7));
draw((0,0)--(9,0)--(9,12)--cycle,linewidth(0.7));
draw((6,8)--(6,0),linewidth(0.7));
draw((5.6,0)--(5.6,0.4)--(6,0.4));
draw((8.6,0)--(8.6,0.4)--(9,0.4));
label("6",(3,0),S);
label("10",(3,4),NW);
label("3",(7.5,0),S);
label("A",(0,0),SW);
label("B",(9,0),SE);
label("C",(9,12),NW);
label("D",(6,0),S);
label("E",(6,8),NW);
[/asy] First of all, we can see that $DE = 8$ since we recognize a $3:4:5$ triangle in $ADE$. Also, thanks to SAS Similarity (or AA also works), we see that $\triangle ABC \sim \triangle ADE.$ That means that $ABC$ is also a $3:4:5$ triangle. Since $AB = 9,$ that means that $AC = 15$ and $BC = 12.$

We find the shaded area of $BCED$ by subtracting the areas of $ABC$ and $ADE.$ The area of $ABC$ is simply $\frac{9 \cdot 12}{2} = 54,$ and the area of $ADE$ is $\frac{6 \cdot 8}{2} = 24.$ Therefore, our desired area is $54 - 24 = \boxed{30} \text{ cm}^2.$

metric geometry - area

Question: What is the number of square centimeters in the shaded area? (The 10 represents the hypotenuse of the white triangle only.) <image1>

Answer: 30


Question:
Points $ A, B, C$ and $ D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

<image1>

Answer: 30

metric geometry - area

Question:
Points $ A, B, C$ and $ D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

<image1>

Answer: 30


Question:
<image1>

Given that all angles shown are marked, the perimeter of the polygon shown is

Choices:
- $14$
- $20$
- $28$
- $48$
- $\text{cannot be determined from the information given}$

metric geometry - length

Question:
<image1>

Given that all angles shown are marked, the perimeter of the polygon shown is
(A) $14$
(B) $20$
(C) $28$
(D) $48$
(E) $\text{cannot be determined from the information given}$


Question:
How many numbers are outside the square?
<image1>

Answer: 2

counting

Question:
How many numbers are outside the square?
<image1>

Answer: 2


Question:
<image1>
How much does Dita weigh?

Choices:
- $2 \mathrm{~kg}$
- $3 \mathrm{~kg}$
- $4 \mathrm{~kg}$
- $5 \mathrm{~kg}$
- $6 \mathrm{~kg}$

algebra

Question:
<image1>
How much does Dita weigh?
(A) $2 \mathrm{~kg}$
(B) $3 \mathrm{~kg}$
(C) $4 \mathrm{~kg}$
(D) $5 \mathrm{~kg}$
(E) $6 \mathrm{~kg}$


Question:
Martin placed 3 different types of objects, hexagons <image1>, squares <image2> and triangles <image3>, on sets of scales, as shown.
<image4>
What does he need to put on the left-hand side on the third set of scales for these scales to balance?

Choices:
- 1 square
- 2 squares
- 1 hexagon
- 1 triangle
- 2 triangles

algebra

Question:
Martin placed 3 different types of objects, hexagons <image1>, squares <image2> and triangles <image3>, on sets of scales, as shown.
<image4>
What does he need to put on the left-hand side on the third set of scales for these scales to balance?
(A) 1 square
(B) 2 squares
(C) 1 hexagon
(D) 1 triangle
(E) 2 triangles


Question:
Three lines disect a big triangle into four triangles and three quadrilaterals. The sum of the perimeters of the three quadrialterals is $25 \mathrm{~cm}$. The sum of the perimeters of the four triangles is $20 \mathrm{~cm}$. The perimeter of the big triangle is $19 \mathrm{~cm}$. How big is the sum of the lengths of the three dissecting lines?
<image1>

Choices:
- $11 \mathrm{~cm}$
- $12 \mathrm{~cm}$
- $13 \mathrm{~cm}$
- $15 \mathrm{~cm}$
- $16 \mathrm{~cm}$

metric geometry - length

Question:
Three lines disect a big triangle into four triangles and three quadrilaterals. The sum of the perimeters of the three quadrialterals is $25 \mathrm{~cm}$. The sum of the perimeters of the four triangles is $20 \mathrm{~cm}$. The perimeter of the big triangle is $19 \mathrm{~cm}$. How big is the sum of the lengths of the three dissecting lines?
<image1>
(A) $11 \mathrm{~cm}$
(B) $12 \mathrm{~cm}$
(C) $13 \mathrm{~cm}$
(D) $15 \mathrm{~cm}$
(E) $16 \mathrm{~cm}$


Question:
<image1>

In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is

Choices:
- $\text{the average of the lengths of the internal and external common tangents}$
- $\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}$
- $\text{always equal to the length of an external common tangent}$
- $\text{greater than the length of an external common tangent}$
- $\text{the geometric mean of the lengths of the internal and external common tangents}$

metric geometry - length

Question:
<image1>

In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is
(A) $\text{the average of the lengths of the internal and external common tangents}$
(B) $\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}$
(C) $\text{always equal to the length of an external common tangent}$
(D) $\text{greater than the length of an external common tangent}$
(E) $\text{the geometric mean of the lengths of the internal and external common tangents}$


Question:
The picture shows a hanging mobile. The mobile weighs 112 grams in total. (The weight of the sticks and threads is not taken into account.) How much does the star weigh?
<image1>

Choices:
- $6 \mathrm{~g}$
- $7 \mathrm{~g}$
- $12 \mathrm{~g}$
- $16 \mathrm{~g}$
- It cannot be calculated.

algebra

Question:
The picture shows a hanging mobile. The mobile weighs 112 grams in total. (The weight of the sticks and threads is not taken into account.) How much does the star weigh?
<image1>
(A) $6 \mathrm{~g}$
(B) $7 \mathrm{~g}$
(C) $12 \mathrm{~g}$
(D) $16 \mathrm{~g}$
(E) It cannot be calculated.


Question:
In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\frac{1}{2}$ inches. The length of $RS$, in inches, is:

<image1>

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

metric geometry - length

Question:
In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\frac{1}{2}$ inches. The length of $RS$, in inches, is:

<image1>
(A) $\text{undetermined}$
(B) $4$
(C) $5\frac{1}{2}$
(D) $6$
(E) $6\frac{1}{4}$


Question:
Vilma took a sheet of paper measuring $10 \mathrm{~cm} \times 20 \mathrm{~cm}$ and made two folds, taking the two smaller sides of the sheet to a diagonal of it. She gets a parallelogram, as shown in the picture. What is the area of this quadrilateral, in $\mathrm{cm}^{2}$?
<image1>

Choices:
- $\frac{100 \sqrt{5}}{3}$
- $50 \sqrt{5}$
- $100(\sqrt{5}-1)$
- $50(5-\sqrt{5})$
- $50(5+\sqrt{5})$

transformation geometry

Question:
Vilma took a sheet of paper measuring $10 \mathrm{~cm} \times 20 \mathrm{~cm}$ and made two folds, taking the two smaller sides of the sheet to a diagonal of it. She gets a parallelogram, as shown in the picture. What is the area of this quadrilateral, in $\mathrm{cm}^{2}$?
<image1>
(A) $\frac{100 \sqrt{5}}{3}$
(B) $50 \sqrt{5}$
(C) $100(\sqrt{5}-1)$
(D) $50(5-\sqrt{5})$
(E) $50(5+\sqrt{5})$


Question:
In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
<image1>

Choices:
- $\frac{1}{2}(\sqrt{6}-2)$
- $\frac{1}{4}$
- $2-\sqrt{3}$
- $\frac{\sqrt{3}}{6}$
- $1-\frac{\sqrt{2}}{2}$

metric geometry - length

Question:
In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
<image1>
(A) $\frac{1}{2}(\sqrt{6}-2)$
(B) $\frac{1}{4}$
(C) $2-\sqrt{3}$
(D) $\frac{\sqrt{3}}{6}$
(E) $1-\frac{\sqrt{2}}{2}$


Question:
The diagram shows a cube with sides of length $12 \mathrm{~cm}$. An ant moves on the cube surface from point $M$ to point $N$ following the route shown. Find the length of ant's path.
<image1>

Choices:
- 40 cm
- 48 cm
- 50 cm
- 60 cm
- It is impossible to determine

solid geometry

Question:
The diagram shows a cube with sides of length $12 \mathrm{~cm}$. An ant moves on the cube surface from point $M$ to point $N$ following the route shown. Find the length of ant's path.
<image1>
(A) 40 cm
(B) 48 cm
(C) 50 cm
(D) 60 cm
(E) It is impossible to determine


Question:
<image1>

Genevieve puts bracing on her large kite in the form of a cross connecting opposite corners of the kite. How many inches of bracing material does she need?

Answer: 39

combinatorial geometry

Question:
<image1>

Genevieve puts bracing on her large kite in the form of a cross connecting opposite corners of the kite. How many inches of bracing material does she need?

Answer: 39


Question:
Figure $OPQR$ is a square.  Point $O$ is the origin, and point $Q$ has coordinates $(2,2)$.  What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?
<image1>

NOT TO SCALE

Choices:
- (-6,0)
- (-4,0)
- (-2,0)
- (2,0)
- (4,0)

analytic geometry

Question:
Figure $OPQR$ is a square.  Point $O$ is the origin, and point $Q$ has coordinates $(2,2)$.  What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?
<image1>

NOT TO SCALE
(A) (-6,0)
(B) (-4,0)
(C) (-2,0)
(D) (2,0)
(E) (4,0)