Question:
Nick wants to write whole numbers into the cells of the $3 \times 3$-table on the right so that the sum of the digits in each in each $2 \times 2$-sub-table is always 10. Five numbers have already been written. Determine the sum of the remaining four numbers.
<image1>

Answer: 12

logic

Question:
Nick wants to write whole numbers into the cells of the $3 \times 3$-table on the right so that the sum of the digits in each in each $2 \times 2$-sub-table is always 10. Five numbers have already been written. Determine the sum of the remaining four numbers.
<image1>

Answer: 12


Question:
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together?

<image1>

Answer: 120

statistics

Question:
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together?

<image1>

Answer: 120


Question:
The cube shown is divided into 64 small cubes. Exactly one of the cubes is grey, as shown in the diagram. Two cubes are said to be 'neighbours' if they have a common face.
On the first day, the white neighbours of the grey cube are changed to grey. On the second day, the white neighbours of all the grey cubes are changed to grey.
How many grey cubes are there at the end of the second day? <image1>

Answer: 17

Solution: The diagram below shows the day on which certain cubes turned grey.  \begin{tabular}{|l|l|l|l|} \hline 2 & 1 & 2 & \\ \hline 1 & 0 & 1 & 2 \\ \hline 2 & 1 & 2 & \\ \hline & 2 & & \\ \hline \multicolumn{5}{|c|}{ top layer } \end{tabular} ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-213.jpg?height=380&width=716&top_left_y=1329&top_left_x=1004)  As can be seen, at the end of the second day there are $11+5+1=17$ grey cubes.

solid geometry

Question:
The cube shown is divided into 64 small cubes. Exactly one of the cubes is grey, as shown in the diagram. Two cubes are said to be 'neighbours' if they have a common face.
On the first day, the white neighbours of the grey cube are changed to grey. On the second day, the white neighbours of all the grey cubes are changed to grey.
How many grey cubes are there at the end of the second day? <image1>

Answer: 17


Question:
These children are standing in a line. Some are facing forwards and others are facing backwards. How many children are holding another child's hand with their right hand?
<image1>

Answer: 6

counting

Question:
These children are standing in a line. Some are facing forwards and others are facing backwards. How many children are holding another child's hand with their right hand?
<image1>

Answer: 6


Question:
Barney has 16 cards: 4 blue $(B), 4$ red $(R), 4$ green $(G)$ and 4 yellow (Y). He wants to place them in the square shown so that every row and every column has exactly one of each card. The diagram shows how he started. How many different ways can he finish?
<image1>

Answer: 4

Solution: The left-hand two columns can only be completed in one way as shown.  \begin{tabular}{|cc|c|} \hline$B$ & $R$ & 1 \\ $R$ & $B$ & 1 \\ \hline$Y$ & $G$ & 2 \\ $G$ & $Y$ & \\ \hline \end{tabular} Then the square labelled 1 can only be $\begin{array}{llll}Y & G & G & Y \\ G & Y\end{array}$ or $\begin{array}{ll}Y & G\end{array}$ and the square labelled 2 can only be $\begin{array}{llll}R & B \\ B & R\end{array}$ or $\begin{array}{lll}B & R \\ R & B\end{array}$. All combinations are possible, giving $2 \times 2$ ways in total.

combinatorics

Question:
Barney has 16 cards: 4 blue $(B), 4$ red $(R), 4$ green $(G)$ and 4 yellow (Y). He wants to place them in the square shown so that every row and every column has exactly one of each card. The diagram shows how he started. How many different ways can he finish?
<image1>

Answer: 4


Question:
Three lines intersect at one point. Two angles are given in the figure. How many degrees does the grey angle have?
<image1>

Choices:
- $52^{\circ}$
- $53^{\circ}$
- $54^{\circ}$
- $55^{\circ}$
- $56^{\circ}$

metric geometry - angle

Question:
Three lines intersect at one point. Two angles are given in the figure. How many degrees does the grey angle have?
<image1>
(A) $52^{\circ}$
(B) $53^{\circ}$
(C) $54^{\circ}$
(D) $55^{\circ}$
(E) $56^{\circ}$


Question:
In triangle $CAT$, we have $\angle ACT = \angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ bisects $\angle ATC$, then $\angle CRT =$

<image1>

Choices:
- $36^\circ$
- $54^\circ$
- $72^\circ$
- $90^\circ$
- $108^\circ$

metric geometry - angle

Question:
In triangle $CAT$, we have $\angle ACT = \angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ bisects $\angle ATC$, then $\angle CRT =$

<image1>
(A) $36^\circ$
(B) $54^\circ$
(C) $72^\circ$
(D) $90^\circ$
(E) $108^\circ$


Question:
During a rough sailing trip, Jacques tried to sketch a map of his village. He managed to draw the four streets, the seven places where they cross and the houses of his friends. The houses are marked on the correct streets, and the intersections are correct, however, in reality, Arrow Street, Nail Street and Ruler Street are all absolutely straight. The fourth street is Curvy Street. Who lives on Curvy Street?
<image1>

Choices:
- Adeline
- Benjamin
- Carole
- David
- It is impossible to tell without a better map

Solution: A pair of straight lines intersects at most once, but Adeline's and Carole's roads intersect twice so one of them must be Curvy Street; similarly Adeline's and Benjamin's roads intersect twice so one of them must also be Curvy Street. Therefore Adeline lives on Curvy Street.

topology

Question:
During a rough sailing trip, Jacques tried to sketch a map of his village. He managed to draw the four streets, the seven places where they cross and the houses of his friends. The houses are marked on the correct streets, and the intersections are correct, however, in reality, Arrow Street, Nail Street and Ruler Street are all absolutely straight. The fourth street is Curvy Street. Who lives on Curvy Street?
<image1>
(A) Adeline
(B) Benjamin
(C) Carole
(D) David
(E) It is impossible to tell without a better map


Question:
The diagram shows a quadrilateral $A B C D$, in which $A D=B C$, $\angle C A D=50^{\circ}, \angle A C D=65^{\circ}$ and $\angle A C B=70^{\circ}$.
<image1>
What is the size of $\angle A B C$ ?

Choices:
- $50^{\circ}$
- $55^{\circ}$
- $60^{\circ}$
- $65^{\circ}$
- Impossible to determine

Solution: From the angle sum of a triangle, $\angle A D C=65^{\circ}$. Since $\angle A D C=\angle A C D$, triangle $A C D$ is isosceles and so $A C=A D=B C$. Triangle $A B C$ is therefore isosceles and from the angle sum of a triangle, $\angle B A C=\angle A B C=55^{\circ}$.

metric geometry - angle

Question:
The diagram shows a quadrilateral $A B C D$, in which $A D=B C$, $\angle C A D=50^{\circ}, \angle A C D=65^{\circ}$ and $\angle A C B=70^{\circ}$.
<image1>
What is the size of $\angle A B C$ ?
(A) $50^{\circ}$
(B) $55^{\circ}$
(C) $60^{\circ}$
(D) $65^{\circ}$
(E) Impossible to determine


Question:
Mr Beaver re-arranges the parts to build a kangaroo.
<image1>
Which part is missing?
<image2>

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

combinatorial geometry

Question:
Mr Beaver re-arranges the parts to build a kangaroo.
<image1>
Which part is missing?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.

<image1>

Answer: 53

solid geometry

Question:
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.

<image1>

Answer: 53


Question: In the diagram, $V W X$ and $X Y Z$ are congruent equilateral triangles and angle $V X Y=80^{\circ}$. What is the size of angle $V W Y$ ? <image1>

Choices:
- $25^{\circ}$
- $30^{\circ}$
- $35^{\circ}$
- $40^{\circ}$
- $45^{\circ}$

Solution: The angles of an equilateral triangle are $60^{\circ}$ and triangle $W X Y$ is isosceles. $\angle W X Y=80^{\circ}+60^{\circ}=140^{\circ}$, hence $\angle X W Y=\frac{1}{2}\left(180^{\circ}-140^{\circ}\right)=20^{\circ}$ and so $\angle V W Y=60^{\circ}-20^{\circ}=40^{\circ}$.

metric geometry - angle

Question:
In the diagram, $V W X$ and $X Y Z$ are congruent equilateral triangles and angle $V X Y=80^{\circ}$. What is the size of angle $V W Y$ ? <image1>
(A) $25^{\circ}$
(B) $30^{\circ}$
(C) $35^{\circ}$
(D) $40^{\circ}$
(E) $45^{\circ}$


Question: Five squares and two right-angled triangles are positioned as shown. The areas of three squares are $3 \mathrm{~m}^{2}, 7 \mathrm{~m}^{2}$ and $22 \mathrm{~m}^{2}$ as shown. What is the area, in $\mathrm{m}^{2}$, of the square with the question mark? <image1>

Answer: 18

Solution: Notice that the central square shares an edge with both of the right-angled triangles and in each case the shared side is the hypotenuse of the triangle. By Pythagoras' Theorem, the area of the central square is equal to the sum of the areas of the squares on the shorter sides. By considering the triangle on the left we see the area, in $\mathrm{m}^{2}$, of the central square is $22+3=25$. Then, by considering the triangle on the right we see that the unknown area, in $\mathrm{m}^{2}$, is $25-7=18$.

metric geometry - area

Question: Five squares and two right-angled triangles are positioned as shown. The areas of three squares are $3 \mathrm{~m}^{2}, 7 \mathrm{~m}^{2}$ and $22 \mathrm{~m}^{2}$ as shown. What is the area, in $\mathrm{m}^{2}$, of the square with the question mark? <image1>

Answer: 18


Question:
In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is 18 and that of the smallest one is 8, then the radius of the middle circle is

<image1>

Answer: 12

metric geometry - length

Question:
In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is 18 and that of the smallest one is 8, then the radius of the middle circle is

<image1>

Answer: 12


Question:
Julia has 2017 round discs available: 1009 black ones and 1008 white ones. Using them, she wants to lay the biggest square pattern (as shown) possible and starts by using a black disc in the left upper corner. Subsequently she lays the discs in such a way that the colours alternate in each row and column. How many discs are left over
when she has laid the biggest square possible?
<image1>

Choices:
- none
- 40 of each colour
- 40 black and 41 white ones
- 41 of each colour
- 40 white and 41 black ones

algebra

Question:
Julia has 2017 round discs available: 1009 black ones and 1008 white ones. Using them, she wants to lay the biggest square pattern (as shown) possible and starts by using a black disc in the left upper corner. Subsequently she lays the discs in such a way that the colours alternate in each row and column. How many discs are left over
when she has laid the biggest square possible?
<image1>
(A) none
(B) 40 of each colour
(C) 40 black and 41 white ones
(D) 41 of each colour
(E) 40 white and 41 black ones


Question: In the figure, square $WXYZ$ has a diagonal of 12 units. Point $A$ is a midpoint of segment $WX$, segment $AB$ is perpendicular to segment $AC$ and $AB = AC.$ What is the length of segment $BC$? <image1>

Answer: 18

Solution:
Triangles WXY and BXY are isosceles triangles that have a leg in common, so they are congruent.  Therefore segment $YB$ is equal to a diagonal of square $WXYZ$, so its length is 12 units. By adding point $D$, as shown, we can see that triangles $CDY$ and $YXB$ are similar to triangle $CAB$. This also means that triangle $CDY$ is similar to triangle $YXB$. Since the sides of two similar triangles are related by a constant factor, and we can see that the length of $DY$ is 1/2 the length of $XB$, we know that the length of $CY$ must be $(1/2)(12) = 6$ units. Thus, the length of CB is $12 + 6 = \boxed{18\text{ units}}$. [asy]
import olympiad; size(150); defaultpen(linewidth(0.8));
draw(unitsquare);
draw((2,0)--(0.5,0)--(0.5,1.5)--cycle);
label("$W$",(0,0),W); label("$X$",(1,0),S); label("$Y$",(1,1),E); label("$Z$",(0,1),W);
label("$A$",(0.5,0),S); label("$B$",(2,0),E); label("$C$",(0.5,1.5),N);
label("$D$",(0.5,1),NW);
[/asy]

metric geometry - length

Question: In the figure, square $WXYZ$ has a diagonal of 12 units. Point $A$ is a midpoint of segment $WX$, segment $AB$ is perpendicular to segment $AC$ and $AB = AC.$ What is the length of segment $BC$? <image1>

Answer: 18


Question:
Bettina chooses five points $A, B, C, D$ and $E$ on a circle and draws the tangent to the circle at point $A$. She realizes that the five angles marked $x$ are all equally big. (Note that the diagram is not drawn to scale!) How big is the angle $\angle A B D$?
<image1>

Choices:
- $66^{\circ}$
- $70.5^{\circ}$
- $72^{\circ}$
- $75^{\circ}$
- $77.5^{\circ}$

metric geometry - angle

Question:
Bettina chooses five points $A, B, C, D$ and $E$ on a circle and draws the tangent to the circle at point $A$. She realizes that the five angles marked $x$ are all equally big. (Note that the diagram is not drawn to scale!) How big is the angle $\angle A B D$?
<image1>
(A) $66^{\circ}$
(B) $70.5^{\circ}$
(C) $72^{\circ}$
(D) $75^{\circ}$
(E) $77.5^{\circ}$


Question:
A panel is composed of 4 circles. When Lucy touches a circle, this circle and the others that touch this circle change their color from white to black or from black to white, as shown in the picture. Starting with all white circles, at least how many circles must Lucy touch, one after the other, so that all circles turn black?
<image1>

Answer: 4

combinatorial geometry

Question:
A panel is composed of 4 circles. When Lucy touches a circle, this circle and the others that touch this circle change their color from white to black or from black to white, as shown in the picture. Starting with all white circles, at least how many circles must Lucy touch, one after the other, so that all circles turn black?
<image1>

Answer: 4


Question:
It takes 9 litres of paint to cover the surface of the cube on the left.
<image1>
How much paint would it take to cover the surface of the shape on the right?

Choices:
- 9 litres
- 8 litres
- 6 litres
- 4 litres
- 2 litres

Solution: The surface areas of the two solids are the same. Hence the same amount of paint is required to cover them. Therefore it would take 9 litres of paint to cover the surface of the second solid.

solid geometry

Question:
It takes 9 litres of paint to cover the surface of the cube on the left.
<image1>
How much paint would it take to cover the surface of the shape on the right?
(A) 9 litres
(B) 8 litres
(C) 6 litres
(D) 4 litres
(E) 2 litres


Question:
A small square is inscribed in a big one as shown in the figure. Find the area of the small square.
<image1>

Answer: 34

metric geometry - area

Question:
A small square is inscribed in a big one as shown in the figure. Find the area of the small square.
<image1>

Answer: 34


Question:
When you put the 4 puzzle pieces together correctly, they form a rectangle with a calculation on it. What is the result of this calculation?
<image1>

Answer: 15

arithmetic

Question:
When you put the 4 puzzle pieces together correctly, they form a rectangle with a calculation on it. What is the result of this calculation?
<image1>

Answer: 15


Question:
Fridolin the hamster runs through the maze shown on the right. On the path there are 16 pumpkin seeds. He is only allowed to cross each junction once. What is the maximum number of pumpkin seeds that he can collect?
<image1>

Answer: 13

graph theory

Question:
Fridolin the hamster runs through the maze shown on the right. On the path there are 16 pumpkin seeds. He is only allowed to cross each junction once. What is the maximum number of pumpkin seeds that he can collect?
<image1>

Answer: 13


Question:
A flee stands on the floor and wants to climb the 10 steps. He can either jump 3 steps upwards or jump 4 steps downwards. What is is the smallest number of jumps he must make?
<image1>

Answer: 8

algebra

Question:
A flee stands on the floor and wants to climb the 10 steps. He can either jump 3 steps upwards or jump 4 steps downwards. What is is the smallest number of jumps he must make?
<image1>

Answer: 8


Question:
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?

<image1>

Answer: 3

analytic geometry

Question:
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?

<image1>

Answer: 3


Question:
The figure shows three concentric circles with four lines passing through their common centre. What percentage of the figure is shaded?
<image1>

Choices:
- $30 \%$
- $35 \%$
- $40 \%$
- $45 \%$
- $50 \%$

metric geometry - area

Question:
The figure shows three concentric circles with four lines passing through their common centre. What percentage of the figure is shaded?
<image1>
(A) $30 \%$
(B) $35 \%$
(C) $40 \%$
(D) $45 \%$
(E) $50 \%$


Question:
The following figures show five paths, indicated by the thickest lines, between the $X$ and $Y$ points. Which of these paths is the longest?
<image1>

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

counting

Question:
The following figures show five paths, indicated by the thickest lines, between the $X$ and $Y$ points. Which of these paths is the longest?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The diameter of the circle from the picture is $10 \mathrm{~cm}$. What is the perimeter of the figure which is marked with double line, if the rectangles in the picture are coincident?
<image1>

Choices:
- $8 \mathrm{~cm}$
- $16 \mathrm{~cm}$
- $20 \mathrm{~cm}$
- $25 \mathrm{~cm}$
- $30 \mathrm{~cm}$

metric geometry - length

Question:
The diameter of the circle from the picture is $10 \mathrm{~cm}$. What is the perimeter of the figure which is marked with double line, if the rectangles in the picture are coincident?
<image1>
(A) $8 \mathrm{~cm}$
(B) $16 \mathrm{~cm}$
(C) $20 \mathrm{~cm}$
(D) $25 \mathrm{~cm}$
(E) $30 \mathrm{~cm}$


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

Answer: -2

analytic geometry

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

Answer: -2


Question:
Bia has five coins as shown beside. She went to the grocery store to buy a fruit, using only three coins, without having to receive change. Among the prices of the following fruits, which one can she NOT buy?
<image1>

Choices:
- 1,30
- 1,35
- 1,40
- 1,55
- 1,75

combinatorics

Question:
Bia has five coins as shown beside. She went to the grocery store to buy a fruit, using only three coins, without having to receive change. Among the prices of the following fruits, which one can she NOT buy?
<image1>
(A) 1,30
(B) 1,35
(C) 1,40
(D) 1,55
(E) 1,75


Question:
In a triangle $A B C$ the points $M$ and $N$ are placed on side $A B$ so that $A N=A C$ and $B M=$ $B C$. Determine $\angle A C B$ if $\angle M C N=43^{\circ}$
<image1>

Choices:
- $86^{\circ}$
- $89^{\circ}$
- $90^{\circ}$
- $92^{\circ}$
- $94^{\circ}$

metric geometry - angle

Question:
In a triangle $A B C$ the points $M$ and $N$ are placed on side $A B$ so that $A N=A C$ and $B M=$ $B C$. Determine $\angle A C B$ if $\angle M C N=43^{\circ}$
<image1>
(A) $86^{\circ}$
(B) $89^{\circ}$
(C) $90^{\circ}$
(D) $92^{\circ}$
(E) $94^{\circ}$


Question:
Ronja had four white tokens and Wanja had four grey tokens. They played a game in which they took turns to place one of their tokens to create two piles. Ronja placed her first token first. Which pair of piles could they not create?
<image1>

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

combinatorics

Question:
Ronja had four white tokens and Wanja had four grey tokens. They played a game in which they took turns to place one of their tokens to create two piles. Ronja placed her first token first. Which pair of piles could they not create?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The fractions $\frac{1}{3}$ and $\frac{1}{5}$ have been placed on the
<image1>
number-line shown on the right. At which position should the fraction $\frac{1}{4}$ be placed?

Choices:
- $a$
- $b$
- $C$
- $d$
- $e$

Solution: The difference between $\frac{1}{3}$ and $\frac{1}{5}$ is $\frac{1}{3}-\frac{1}{5}=\frac{2}{15}$. This section of the number line is divided into 16 intervals, each of length $\frac{2}{15} \div 16=\frac{1}{120}$. The difference between $\frac{1}{4}$ and $\frac{1}{5}$ is $\frac{1}{4}-\frac{1}{5}=\frac{1}{20}=\frac{6}{120}$, and hence $\frac{1}{4}$ is six smaller intervals from $\frac{1}{5}$, at point $a$.

algebra

Question:
The fractions $\frac{1}{3}$ and $\frac{1}{5}$ have been placed on the
<image1>
number-line shown on the right. At which position should the fraction $\frac{1}{4}$ be placed?
(A) $a$
(B) $b$
(C) $C$
(D) $d$
(E) $e$


Question: Barbara wants to complete the diagram below by inserting three numbers, one into each empty cell. She wants the sum of the first three numbers to be 100 , the sum of the middle three numbers to be 200 and the sum of the last three numbers to be 300 . What number should Barbara insert into the middle cell of the diagram? <image1>

Answer: 60

Solution: \begin{tabular}{|l|l|l|l|l|} \hline 10 & $X$ & $Y$ & $Z$ & 130 \\ \hline \end{tabular}  If we label the values in the cells as shown, the question tells us that $10+X+Y=100$, $X+Y+Z=200$ and $Y+Z+130=300$. The first two equations give $Z=110$ and substituting this into the third equation then gives $Y=60$.

algebra

Question: Barbara wants to complete the diagram below by inserting three numbers, one into each empty cell. She wants the sum of the first three numbers to be 100 , the sum of the middle three numbers to be 200 and the sum of the last three numbers to be 300 . What number should Barbara insert into the middle cell of the diagram? <image1>

Answer: 60


Question: The diagram shows a grey rectangle that lies within a bigger rectangle which sides it touches. Two corner points of the grey rectangle are the midpoints of the shorter sides of the bigger rectangle. The grey rectangle is made up of three squares that each have an area of $25 \mathrm{~cm}^{2}$. How big is the area of the bigger rectangle in $\mathrm{cm}^{2}$ ? <image1>

Answer: 150

metric geometry - area

Question: The diagram shows a grey rectangle that lies within a bigger rectangle which sides it touches. Two corner points of the grey rectangle are the midpoints of the shorter sides of the bigger rectangle. The grey rectangle is made up of three squares that each have an area of $25 \mathrm{~cm}^{2}$. How big is the area of the bigger rectangle in $\mathrm{cm}^{2}$ ? <image1>

Answer: 150


Question:
In the sum shown, different shapes represent different digits.
<image1>
What digit does the square represent?

Answer: 6

Solution: The maximum value one can obtain by adding a two-digit number and two one-digit numbers is $99+9+9=117$. Hence the triangle must represent 1 and therefore the sum of the three numbers is 111 . To obtain this answer to the sum, the circle must represent 9 since $89+9+9$ is less than 111 . Hence the two squares must sum to $111-99=12$. Therefore the square represents the digit 6 .

algebra

Question:
In the sum shown, different shapes represent different digits.
<image1>
What digit does the square represent?

Answer: 6


Question:
Two of the following four facts about a positive integer $N$ are true and two are false. <image1>
What is the value of $N$ ?

Answer: 5

Solution: If $N$ were divisible by 55 , then it would also be divisible by 5 and 11 , making three statements true. Hence $N$ is not divisible by 55 . Therefore exactly two of the remaining statements are true. It is not possible for $N$ to be both less than 10 and divisible by 11, and it is not possible for $N$ to be divisible by both 5 and 11 without also being divisible by 55 . Therefore the two true statements are $N$ is divisible by 5 and $N$ is less than 10 . Hence the value of $N$ is 5 .

logic

Question:
Two of the following four facts about a positive integer $N$ are true and two are false. <image1>
What is the value of $N$ ?

Answer: 5


Question:
The curve in the diagram is defined by the equation
$$
\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right)
$$
Which of the lines $a, b, c, d$ is the $y$-axis?
<image1>

Choices:
- $a$
- $b$
- $c$
- $d$
- none of them

analytic geometry

Question:
The curve in the diagram is defined by the equation
$$
\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right)
$$
Which of the lines $a, b, c, d$ is the $y$-axis?
<image1>
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) none of them


Question:
<image1>

If the sum of the measures in degrees of angles $A,~B,~C,~D,~E$ and $F$ in the figure above is $90n$, then $n$ is equal to

Answer: 4

metric geometry - angle

Question:
<image1>

If the sum of the measures in degrees of angles $A,~B,~C,~D,~E$ and $F$ in the figure above is $90n$, then $n$ is equal to

Answer: 4


Question: The volume of the cylinder shown is $45\pi$ cubic cm. What is the height in centimeters of the cylinder? <image1>

Answer: 5

Solution: The volume of the cylinder is $bh=\pi r^2h$. The radius of the base is $3$ cm, so we have $9\pi h=45\pi\qquad\Rightarrow h=5$. The height of the cylinder is $\boxed{5}$ cm.

solid geometry

Question: The volume of the cylinder shown is $45\pi$ cubic cm. What is the height in centimeters of the cylinder? <image1>

Answer: 5


Question:
The diagram shows squares of different sizes. The side length of the smallest square is $20 \mathrm{~cm}$. How long is the black line?
<image1>

Choices:
- $380 \mathrm{~cm}$
- $400 \mathrm{~cm}$
- $420 \mathrm{~cm}$
- $440 \mathrm{~cm}$
- $1680 \mathrm{~cm}$

combinatorial geometry

Question:
The diagram shows squares of different sizes. The side length of the smallest square is $20 \mathrm{~cm}$. How long is the black line?
<image1>
(A) $380 \mathrm{~cm}$
(B) $400 \mathrm{~cm}$
(C) $420 \mathrm{~cm}$
(D) $440 \mathrm{~cm}$
(E) $1680 \mathrm{~cm}$


Question:
Lina has already laid two shapes on a square playing board. Which of the 5 shapes can she add to the board so that none of the remaining four shapes will have space to fit.
<image1>
<image2>

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

combinatorial geometry

Question:
Lina has already laid two shapes on a square playing board. Which of the 5 shapes can she add to the board so that none of the remaining four shapes will have space to fit.
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the picture $A B C D$ is a square of side 1 and the semicircles have centers on $A, B, C$ and $D$. What is the length of $P Q$?
<image1>

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

metric geometry - length

Question:
In the picture $A B C D$ is a square of side 1 and the semicircles have centers on $A, B, C$ and $D$. What is the length of $P Q$?
<image1>
(A) $2-\sqrt{2}$
(B) $\frac{3}{4}$
(C) $\sqrt{5}-\sqrt{2}$
(D) $\frac{\sqrt{3}}{3}$
(E) $\sqrt{3}-1$


Question:
Five cars are labelled with the numbers 1 to 5 . They drive in the direction of the arrow.
<image1>
First the last car overtakes the two cars in front of it.
Then the now second to last car overtakes the two in front of it.
In the end the car that is now in the middle overtakes the two in front of it.
In which order do the cars now drive?

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

combinatorial geometry

Question:
Five cars are labelled with the numbers 1 to 5 . They drive in the direction of the arrow.
<image1>
First the last car overtakes the two cars in front of it.
Then the now second to last car overtakes the two in front of it.
In the end the car that is now in the middle overtakes the two in front of it.
In which order do the cars now drive?
(A) $1,2,3,4,5$
(B) $2,1,3,5,4$
(C) $2,1,5,3,4$
(D) $3,1,4,2,5$
(E) $4,1,2,5,3$


Question:
There are 10 kangaroos in a row, as seen in the picture. Two kangaroos, that are standing next to each other and can see each other are allowed to change places by hopping past each other. This is carried out until no more jumps are allowed. How often do two kangaroos swap places?
<image1>

Answer: 18

counting

Question:
There are 10 kangaroos in a row, as seen in the picture. Two kangaroos, that are standing next to each other and can see each other are allowed to change places by hopping past each other. This is carried out until no more jumps are allowed. How often do two kangaroos swap places?
<image1>

Answer: 18


Question:
The diagram shows a rectangle $A B E F$ and a triangle $A B C$. We know that the angle $A C F$ equals angle $C B E$. If $F C=6$ and $C E=2$ then the area of $A B C$ is:
<image1>

Choices:
- 12
- 16
- $8 \sqrt{2}$
- $8 \sqrt{3}$
- Another value

metric geometry - area

Question:
The diagram shows a rectangle $A B E F$ and a triangle $A B C$. We know that the angle $A C F$ equals angle $C B E$. If $F C=6$ and $C E=2$ then the area of $A B C$ is:
<image1>
(A) 12
(B) 16
(C) $8 \sqrt{2}$
(D) $8 \sqrt{3}$
(E) Another value


Question:
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares.
<image1>
The measure of $\angle GDA$ is

Choices:
- $90^\circ$
- $105^\circ$
- $120^\circ$
- $135^\circ$
- $150^\circ$

metric geometry - angle

Question:
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares.
<image1>
The measure of $\angle GDA$ is
(A) $90^\circ$
(B) $105^\circ$
(C) $120^\circ$
(D) $135^\circ$
(E) $150^\circ$


Question:
Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<image1>

Choices:
- $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
- $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right) \neq \operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
- $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)>\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
- $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)<\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
- $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$

solid geometry

Question:
Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<image1>
(A) $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(B) $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right) \neq \operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(C) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)>\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(D) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)<\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(E) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$


Question: Nine steps of a staircase arranged in a cylindrical order starting at the bottom and leading all the way to the top can be seen. All steps are equally high. How many steps cannot be seen? <image1>

Answer: 12

solid geometry

Question: Nine steps of a staircase arranged in a cylindrical order starting at the bottom and leading all the way to the top can be seen. All steps are equally high. How many steps cannot be seen? <image1>

Answer: 12


Question:
Sylvia draws patterns with hexagons as shown. If she carries on drawing in this way, how many hexagons will there be in the fifth pattern?
<image1>

Answer: 61

algebra

Question:
Sylvia draws patterns with hexagons as shown. If she carries on drawing in this way, how many hexagons will there be in the fifth pattern?
<image1>

Answer: 61


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

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

arithmetic

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


Question:
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?

<image1>

Answer: 5

transformation geometry

Question:
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?

<image1>

Answer: 5


Question:
Tom encodes words using the board shown. For example, the word PIZZA has the code $A 2 A 4 C 1 C 1 B 2$. What word did Tom encode as B3B2C4D2?
<image1>

Choices:
- MAZE
- MASK
- MILK
- MATE
- MATH

algebra

Question:
Tom encodes words using the board shown. For example, the word PIZZA has the code $A 2 A 4 C 1 C 1 B 2$. What word did Tom encode as B3B2C4D2?
<image1>
(A) MAZE
(B) MASK
(C) MILK
(D) MATE
(E) MATH


Question:
We consider a $5 \times 5$ square that is split up into 25 fields. Initially all fields are white. In each move it is allowed to change the colour of three fields that are adjacent in a horizontal or vertical line (i.e. white fields turn black and black ones turn white). What is the smallest number of moves needed to obtain the chessboard colouring shown in the diagram?
<image1>

Choices:
- less than 10
- 10
- 12
- more than 12
- This colouring cannot be obtained.

graph theory

Question:
We consider a $5 \times 5$ square that is split up into 25 fields. Initially all fields are white. In each move it is allowed to change the colour of three fields that are adjacent in a horizontal or vertical line (i.e. white fields turn black and black ones turn white). What is the smallest number of moves needed to obtain the chessboard colouring shown in the diagram?
<image1>
(A) less than 10
(B) 10
(C) 12
(D) more than 12
(E) This colouring cannot be obtained.


Question:
Two circles are tangent to each other and also to two sides of a square. What is the measure of the $A \hat{O} B$ angle, determined by three of these points of tangency, as shown in the figure?
<image1>

Choices:
- $110^{\circ}$
- $112^{\circ}$
- $120^{\circ}$
- $128^{\circ}$
- $135^{\circ}$

metric geometry - angle

Question:
Two circles are tangent to each other and also to two sides of a square. What is the measure of the $A \hat{O} B$ angle, determined by three of these points of tangency, as shown in the figure?
<image1>
(A) $110^{\circ}$
(B) $112^{\circ}$
(C) $120^{\circ}$
(D) $128^{\circ}$
(E) $135^{\circ}$


Question:
A goldsmith has 12 double-links of chain. Out of these he wants to make a single closed chain with 24 links. What is the minimum number of links that he must open (and close again)?
<image1>

Answer: 8

arithmetic

Question:
A goldsmith has 12 double-links of chain. Out of these he wants to make a single closed chain with 24 links. What is the minimum number of links that he must open (and close again)?
<image1>

Answer: 8


Question:
The cells of the $4 \times 4$-table on the right should be coloured either in black or white. The numbers determine how many cells in each row/column should be black. How many ways are there to do the colouring in?
<image1>

Answer: 5

logic

Question:
The cells of the $4 \times 4$-table on the right should be coloured either in black or white. The numbers determine how many cells in each row/column should be black. How many ways are there to do the colouring in?
<image1>

Answer: 5


Question:
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$.  Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$.  If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles.  Find the volume of this pyramid.

<image1>

Answer: 594

solid geometry

Question:
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$.  Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$.  If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles.  Find the volume of this pyramid.

<image1>

Answer: 594


Question: In a supermarket trolley park, there are two lines of tightly-packed trolleys. The first line has ten trolleys and is $2.9 \mathrm{~m}$ long. The second line has twenty trolleys and is $4.9 \mathrm{~m}$ long. What is the length of one trolley, in $\mathrm{m}$ ? <image1>

Answer: 1.1

Solution: There are 10 more trolleys in the second line, which adds $2 \mathrm{~m}$ to the length, so each trolley adds $0.2 \mathrm{~m}$. If we subtract nine of these extra lengths from the first line, we will be left with the length of one trolley, namely $2.9 \mathrm{~m}-9 \times 0.2 \mathrm{~m}=1.1 \mathrm{~m}$.

algebra

Question: In a supermarket trolley park, there are two lines of tightly-packed trolleys. The first line has ten trolleys and is $2.9 \mathrm{~m}$ long. The second line has twenty trolleys and is $4.9 \mathrm{~m}$ long. What is the length of one trolley, in $\mathrm{m}$ ? <image1>

Answer: 1.1


Question:
The figure of side 1 is formed by six equal triangles, made with 12 sticks. How many matchsticks are needed to complete the figure of side 2, partially represented?
<image1>

Answer: 36

combinatorial geometry

Question:
The figure of side 1 is formed by six equal triangles, made with 12 sticks. How many matchsticks are needed to complete the figure of side 2, partially represented?
<image1>

Answer: 36


Question:
Ana draws some shapes on a sheet. Her drawing has fewer squares than triangles. What could be her drawing?
<image1>

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

counting

Question:
Ana draws some shapes on a sheet. Her drawing has fewer squares than triangles. What could be her drawing?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The diagram shows a net of an unfolded rectangular box. What is the volume of the box (in $\mathrm{cm}^{3}$ )? <image1>

Answer: 80

Solution: Let the dimensions of the box be $x \mathrm{~cm}$ by $y \mathrm{~cm}$ by $z \mathrm{~cm}$ as indicated. From the diagram, we have $2 x+2 y=26$, $x+z=10$ and $y+z=7$. When we add the last two of these, we obtain $x+y+2 z=17$ and, when we then double this, we obtain $2 x+2 y+4 z=34$. Therefore $4 z=34-26=8$ and hence $z=2$. Therefore $x=8$ and $y=5$ and hence the volume of the box in $\mathrm{cm}^{3}$ is $8 \times 5 \times 2=80$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-245.jpg?height=222&width=494&top_left_y=817&top_left_x=1341)

solid geometry

Question: The diagram shows a net of an unfolded rectangular box. What is the volume of the box (in $\mathrm{cm}^{3}$ )? <image1>

Answer: 80


Question:
Ria wants to write a number into each box. She has already written two numbers. The sum of all five numbers should be 35, the sum of the first three numbers should be 22, the sum of the last three numbers should be 25. What is the product Ria gets, if she multiplies the two numbers in the grey boxes?
<image1>

Answer: 63

algebra

Question:
Ria wants to write a number into each box. She has already written two numbers. The sum of all five numbers should be 35, the sum of the first three numbers should be 22, the sum of the last three numbers should be 25. What is the product Ria gets, if she multiplies the two numbers in the grey boxes?
<image1>

Answer: 63


Question: Four diagonals of a regular octagon with side length 2 intersect as shown.  Find the area of the shaded region. <image1>

Answer: 4\sqrt{2}

Solution:
Label the points as shown below: [asy]
pair A, B, C, D, E, F, G, H;
real x = 22.5;
pair A = dir(x);
pair B = dir(45+x);
pair C = dir(45*2+x);
pair D = dir(45*3+x);
pair E = dir(45*4+x);
pair F = dir(45*5+x);
pair G = dir(45*6+x);
pair H = dir(45*7+x);
draw(A--B--C--D--E--F--G--H--cycle);
label("$A$", A, NE);
label("$B$", B, N);
label("$C$", C, N);
label("$D$", D, NW);
label("$E$", E, SW);
label("$F$", F, S);
label("$G$", G, S);
label("$H$", H, SE);
draw(A--D--G);
draw(C--H--E);
pair Q = intersectionpoint(A--D,C--H);
label("$Q$",Q,NE);
pair R = intersectionpoint(H--E,D--G);
label("$R$",R,NE);
[/asy] We can find the area of $DQHR$ by finding the length of the height and base.  The length of the height is equal to the side length of the octagon, which is 2.  To find the length of base $RH$, we notice that $RH=EH-ER$.  Because of the parallel lines, $ERGF$ is a parallelogram, and thus $ER=FG=2$.

[asy]
draw((0,0)--(2,0)--(2+sqrt(2),sqrt(2))--(2,sqrt(2))--(0,sqrt(2))--(-sqrt(2),sqrt(2))--cycle);
draw((0,0)--(0,sqrt(2)),dashed); draw((2,0)--(2,sqrt(2)),dashed);
label("$F$",(0,0) ,SW ); label("$G$", (2,0), SE); label("$H$",(2+sqrt(2),sqrt(2)) ,NE ); label("$N$",(2,sqrt(2)) ,N ); label("$M$", (0,sqrt(2)),N ); label("$E$",(-sqrt(2),sqrt(2)) ,NW );

[/asy]

To find $EH$, we drop two perpendiculars from $F$ and $G$ to $EH$, creating two isosceles right triangles $\triangle EMF$ and $\triangle HNG$, and one rectangle $MNGF$.  Since we have $EF=FG=GH=2$, we have $MN=2$ as well.  Also, we have $EM=NH=2/\sqrt{2}=\sqrt{2}$.  Thus, $EH=\sqrt{2}+2+\sqrt{2}=2+2\sqrt{2}$.

Finally, we have $RH=EH-ER = 2+2\sqrt{2}-2=2\sqrt{2}$.  The area of parallelogram $DQRH$ is thus $(2\sqrt{2})(2) = \boxed{4\sqrt{2}}$.

metric geometry - area

Question: Four diagonals of a regular octagon with side length 2 intersect as shown.  Find the area of the shaded region. <image1>

Answer: 4\sqrt{2}


Question: A square piece of paper is cut into six rectangular pieces as shown in the diagram. When the lengths of the perimeters of the six rectangular pieces are added together, the result is $120 \mathrm{~cm}$. What is the area of the square piece of paper? <image1>

Choices:
- $48 \mathrm{~cm}^{2}$
- $64 \mathrm{~cm}^{2}$
- $110.25 \mathrm{~cm}^{2}$
- $144 \mathrm{~cm}^{2}$
- $256 \mathrm{~cm}^{2}$

Solution: Let the square have side $x \mathrm{~cm}$. Label each rectangle and its sides as shown where the units are $\mathrm{cm}$. The perimeters are: \begin{tabular}{lll} $A$ & $B$ & $C$ \\ $2(a+b)$ & $2(a+c)$ & $2(a+d)$ \\ $D$ & $E$ & $F$ \\ $2(e+h)$ & $2(f+h)$ & $2(g+h)$ \end{tabular} Thus the total of the perimeters is: $$ \begin{aligned} 2(a+b+a+c+a+d+e+h+f+h+g+h)= \\ 2(3(a+h)+(b+c+d)+(e+f+g)) . \end{aligned} $$ But $a+h=b+c+d=e+f+g=x$. Thus the perimeter is $2(3 x+x+x)=10 x$. So $x=12$ which means that the area of the paper is $144 \mathrm{~cm}^{2}$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-130.jpg?height=390&width=385&top_left_y=626&top_left_x=1445)

metric geometry - area

Question:
A square piece of paper is cut into six rectangular pieces as shown in the diagram. When the lengths of the perimeters of the six rectangular pieces are added together, the result is $120 \mathrm{~cm}$. What is the area of the square piece of paper? <image1>
(A) $48 \mathrm{~cm}^{2}$
(B) $64 \mathrm{~cm}^{2}$
(C) $110.25 \mathrm{~cm}^{2}$
(D) $144 \mathrm{~cm}^{2}$
(E) $256 \mathrm{~cm}^{2}$


Question:
Jörg is sorting his socks. Two socks with the same number are one pair.
<image1>
How many pairs can he find?

Answer: 5

counting

Question:
Jörg is sorting his socks. Two socks with the same number are one pair.
<image1>
How many pairs can he find?

Answer: 5


Question:
The ratio of the radii of two concentric circles is $1:3$.  If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is

<image1>

Answer: 18

metric geometry - length

Question:
The ratio of the radii of two concentric circles is $1:3$.  If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is

<image1>

Answer: 18


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

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

topology

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


Question: The diagram (which $\underline{\text { is }}$ drawn to scale) shows two triangles. In how many ways can you choose two vertices, one in each triangle, so that the straight line through the two vertices does not cross either triangle? <image1>

Answer: 4

Solution: Consider each vertex of the left-hand triangle in turn.  From vertex $P$, no line can be drawn.  From vertex $Q$, a line can be drawn to vertex $X$ and vertex $Y$.  From vertex $R$, a line can be drawn to vertex $X$ and vertex $Z$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-146.jpg?height=214&width=414&top_left_y=944&top_left_x=1455)  Therefore, the two vertices can be chosen in only four ways.

combinatorics

Question: The diagram (which $\underline{\text { is }}$ drawn to scale) shows two triangles. In how many ways can you choose two vertices, one in each triangle, so that the straight line through the two vertices does not cross either triangle? <image1>

Answer: 4


Question:
A square bit of paper is folded along the dashed lines in some order and direction. One of the corners of the resulting small square is cut off. The piece of paper is then unfolded. How many holes are on the inner area of the piece of paper?
<image1>

Answer: 1

transformation geometry

Question:
A square bit of paper is folded along the dashed lines in some order and direction. One of the corners of the resulting small square is cut off. The piece of paper is then unfolded. How many holes are on the inner area of the piece of paper?
<image1>

Answer: 1


Question:
What fraction of the square is shaded?

<image1>

Choices:
- $\frac{1}{3}$
- $\frac{2}{5}$
- $\frac{5}{12}$
- $\frac{3}{7}$
- $\frac{1}{2}$

metric geometry - area

Question:
What fraction of the square is shaded?

<image1>
(A) $\frac{1}{3}$
(B) $\frac{2}{5}$
(C) $\frac{5}{12}$
(D) $\frac{3}{7}$
(E) $\frac{1}{2}$


Question: The diagram shows a net of a cube, with three dotted lines added. If you folded the net into a cube and then cut along the dotted lines you would have a hole in the cube. What would be the shape of the hole? <image1>

Choices:
- an equilateral triangle
- a rectangle, but not a square
- a right-angled triangle
- a square
- a hexagon

Solution: The three dotted lines are equal in length, so the hole has three equal sides. The diagram shows the assembled cube with the hole removed.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-023.jpg?height=286&width=257&top_left_y=1028&top_left_x=1571)

descriptive geometry

Question:
The diagram shows a net of a cube, with three dotted lines added. If you folded the net into a cube and then cut along the dotted lines you would have a hole in the cube. What would be the shape of the hole? <image1>
(A) an equilateral triangle
(B) a rectangle, but not a square
(C) a right-angled triangle
(D) a square
(E) a hexagon


Question:
The fractions $\frac{1}{3}$ und $\frac{1}{5}$ are shown on the number line. In which position should $\frac{1}{4}$ be shown?
<image1>

Choices:
- a
- b
- c
- d
- e

algebra

Question:
The fractions $\frac{1}{3}$ und $\frac{1}{5}$ are shown on the number line. In which position should $\frac{1}{4}$ be shown?
<image1>
(A) a
(B) b
(C) c
(D) d
(E) e


Question: A square has a side length of 10 inches. Congruent isosceles right triangles are cut off each corner so that the resulting octagon has equal side lengths. How many inches are in the length of one side of the octagon? Express your answer as a decimal to the nearest hundredth. <image1>

Answer: 4.14

Solution:
Recall that the ratio of the hypotenuse to the leg of an isosceles right triangle is $\sqrt{2}$.  Looking at the removed triangles, we see that the three segments that make up the side of square are $s/\sqrt{2}$, $s$, and $s/\sqrt{2}$, where $s$ is the side length of the octagon.  Setting the sum of these three side lengths equal to 10 inches, we find \begin{align*}
\frac{s}{\sqrt{2}}+s+\frac{s}{\sqrt{2}}&=10 \implies \\
s\left(\frac{1}{\sqrt{2}}+1+\frac{1}{\sqrt{2}}\right)&=10 \implies \\
s(\sqrt{2}+1)&=10 \implies \\
s&=\frac{10}{\sqrt{2}+1}=10(\sqrt{2}-1),
\end{align*} where we have rationalized the denominator twice:  \[
\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2},
\] and  \[
\frac{10}{\sqrt{2}+1}=\frac{10}{\sqrt{2}+1}\cdot\frac{\sqrt{2}-1}{\sqrt{2}-1}=10(\sqrt{2}-1).
\] To the nearest hundredth, $s=\boxed{4.14}$ inches.

[asy]
size(150);
defaultpen(linewidth(0.7)+fontsize(10));
real s = 10/(1+sqrt(2));
draw((0,0)--(10,0)--(10,10)--(0,10)--(0,0));
draw((5*sqrt(2)/(1 + sqrt(2)),0)--(0,5*sqrt(2)/(1 + sqrt(2))));
draw((10 - (5*sqrt(2)/(1 + sqrt(2))),0)--(10,5*sqrt(2)/(1 + sqrt(2))));
draw((0,(10-(5*sqrt(2)/(1+sqrt(2)))))--(5*sqrt(2)/(1 + sqrt(2)),10));
draw((10 - (5*sqrt(2)/(1 + sqrt(2))),10)--(10,10 - 5*sqrt(2)/(1 + sqrt(2))));

label("$s$",(10-s/(2*sqrt(2)),10-s/(2*sqrt(2))),SW);
label("$\displaystyle{\frac{s}{\sqrt{2}}}$",(10,10-s/(2*sqrt(2))),E);
label("$\displaystyle{\frac{s}{\sqrt{2}}}$",(10,s/(2*sqrt(2))),E);
label("$s$",(10,5),E);

draw(rightanglemark((10,0),(10,10),(0,10)));[/asy]

metric geometry - length

Question: A square has a side length of 10 inches. Congruent isosceles right triangles are cut off each corner so that the resulting octagon has equal side lengths. How many inches are in the length of one side of the octagon? Express your answer as a decimal to the nearest hundredth. <image1>

Answer: 4.14


Question:
A big square is made from 25 small squares put together. A few of the small squares have been lost. How many have been lost?
<image1>

Answer: 10

counting

Question:
A big square is made from 25 small squares put together. A few of the small squares have been lost. How many have been lost?
<image1>

Answer: 10


Question:
Five identical measuring jugs are filled with water. Four of them contain exactly the same amount of water. Which measuring jug contains a different amount?
<image1>

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

statistics

Question:
Five identical measuring jugs are filled with water. Four of them contain exactly the same amount of water. Which measuring jug contains a different amount?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
How many ropes can you see in this picture?
<image1>

Answer: 3

topology

Question:
How many ropes can you see in this picture?
<image1>

Answer: 3


Question:
A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is
<image1>

Answer: 19

solid geometry

Question:
A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is
<image1>

Answer: 19


Question:
Ellen wants to colour some of the cells of a $4 \times 4$ grid. She wants to do this so that each coloured cell shares at least one side with an uncoloured cell and each uncoloured cell shares at least one side with a coloured cell.
What is the largest number of cells she can colour?
<image1>

Answer: 12

Solution: Consider the four cells in the top left corner. It is not possible for all four cells to be coloured or the top left cell would not be touching an uncoloured cell and so there is at least one uncoloured cell in that group of four cells. By a similar argument, there is at least one uncoloured cell amongst the four cells in the bottom left corner,  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-231.jpg?height=217&width=229&top_left_y=1645&top_left_x=1573) amongst the four cells in the bottom right corner and amongst the four cells in the top right corner. Therefore there are at least four uncoloured cells in the grid and hence at most twelve coloured cells. The diagram above shows that an acceptable arrangement is possible with twelve coloured cells.  Hence the largest number of cells Ellen can colour is twelve.

graph theory

Question:
Ellen wants to colour some of the cells of a $4 \times 4$ grid. She wants to do this so that each coloured cell shares at least one side with an uncoloured cell and each uncoloured cell shares at least one side with a coloured cell.
What is the largest number of cells she can colour?
<image1>

Answer: 12


Question:
Black and white tiles can be laid on square floors as shown in the pictures. We can see floors with 4 black and 9 black tiles respectively. In each corner there is a black tile, and each black tile touches only white tiles. How many white tiles would there be on a floor that had 25 black tiles?
<image1>

Answer: 56

algebra

Question:
Black and white tiles can be laid on square floors as shown in the pictures. We can see floors with 4 black and 9 black tiles respectively. In each corner there is a black tile, and each black tile touches only white tiles. How many white tiles would there be on a floor that had 25 black tiles?
<image1>

Answer: 56


Question:
The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7$. How many of these four numbers are prime?
<image1>

Answer: 3

algebra

Question:
The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7$. How many of these four numbers are prime?
<image1>

Answer: 3


Question:
Four cars drive into a roundabout at the same point in time, each one coming from a different direction (see diagram). No car drives all the way around the roundabout, and no two cars leave at the same exit. In how many different ways can the cars exit the roundabout?
<image1>

Answer: 9

combinatorics

Question:
Four cars drive into a roundabout at the same point in time, each one coming from a different direction (see diagram). No car drives all the way around the roundabout, and no two cars leave at the same exit. In how many different ways can the cars exit the roundabout?
<image1>

Answer: 9


Question:
Tatiana's teacher drew a $3 \times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the right (some of the numbers in the cells have been rubbed out.)
<image1>
What number should be in the cell with the question mark?

Answer: 16

Solution: Let $P$ be the number of times the top left $2 \times 2$ square was picked; and let $Q, R, S$ be the corresponding numbers for the top right, bottom right and bottom left $2 \times 2$ squares. The cells in the four corners of the $3 \times 3$ grid each appear in exactly one of these $2 \times 2$ squares, so $S=13$ and the number in the cell marked with the question mark is $R$. The middle top cell is increased $P+Q$ times since it belongs to the top left $2 \times 2$ square and the top right $2 \times 2$ square. So $P+Q=18$. The central square belongs to all four $2 \times 2$ squares, so $47=P+Q+R+S=18+R+13=R+31$. Hence $R=16$.

algebra

Question:
Tatiana's teacher drew a $3 \times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the right (some of the numbers in the cells have been rubbed out.)
<image1>
What number should be in the cell with the question mark?

Answer: 16


Question:
Arno lays out the word KANGAROO using 8 cards. However, some cards are turned.
<image1>
By turning it twice the letter $\mathrm{K}$ can be corrected, letter $\mathrm{A}$ can be corrected by turning it once (see drawing). How often does he have to turn so that the word KANGAROO can be read correctly?
<image2>

Answer: 6

arithmetic

Question:
Arno lays out the word KANGAROO using 8 cards. However, some cards are turned.
<image1>
By turning it twice the letter $\mathrm{K}$ can be corrected, letter $\mathrm{A}$ can be corrected by turning it once (see drawing). How often does he have to turn so that the word KANGAROO can be read correctly?
<image2>

Answer: 6


Question: A piece of paper in the shape of a regular hexagon, as shown, is folded so that the three marked vertices meet at the centre $O$ of the hexagon. What is the shape of the figure that is formed? <image1>

Choices:
- Six-pointed star
- Dodecagon
- Hexagon
- Square
- Equilateral Triangle

Solution: Label vertices $A, B, C, D, E, E$ and $F$ as shown. Since the hexagon is regular, it can be divided into six equilateral triangles as shown. Therefore quadrilateral $O A B C$ is a rhombus and hence its diagonal $A C$ is a line of symmetry. Therefore, if vertex $B$ is folded onto $O$, the fold will be along $A C$. Similarly, if vertices $D$ and $F$ are folded onto $O$, the  ![](https://cdn.mathpix.com/cropped/2023_12_27_9b94075f4aebf6f0284ag-1.jpg?height=277&width=309&top_left_y=2077&top_left_x=1459) folds will be along $C E$ and $E A$ respectively. Hence the figure that is formed will be a triangle and, since all three of the rhombuses $O A B C, O C D E$ and $O E F A$ are made out of two congruent equilateral triangles, the lengths of their diagonals $A C, C E$ and $E A$ will be equal. Hence the shape $A C E$ that is formed is an equilateral triangle.

transformation geometry

Question:
A piece of paper in the shape of a regular hexagon, as shown, is folded so that the three marked vertices meet at the centre $O$ of the hexagon. What is the shape of the figure that is formed? <image1>
(A) Six-pointed star
(B) Dodecagon
(C) Hexagon
(D) Square
(E) Equilateral Triangle


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

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

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

transformation geometry

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


Question:
<image1>

In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$.

Choices:
- $10^\circ$
- $15^\circ$
- $20^\circ$
- $\left(\frac{45}{2}\right)^\circ$
- $30^\circ$

metric geometry - angle

Question:
<image1>

In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$.
(A) $10^\circ$
(B) $15^\circ$
(C) $20^\circ$
(D) $\left(\frac{45}{2}\right)^\circ$
(E) $30^\circ$


Question:
In the calculation alongside, different letters represent different digits.
<image1>
Find the least possible answer to the subtraction shown.

Answer: 110

Solution: $\mathrm{K}$ and $\mathrm{R}$ are hundreds so must be as large as possible, i.e. 9 or 8 (in either order). $\mathrm{A}, \mathrm{G}, \mathrm{O}$ are all in the tens column so they should be 7,6,5, but $\mathrm{A}$ and $\mathrm{O}$ are repeated in the units so should be maximised to 7 and 6 (in either order), leaving $\mathrm{G}=5 . \mathrm{N}$ must be 4 . One possible minimal result is $2007-974-57-866=110$.

algebra

Question:
In the calculation alongside, different letters represent different digits.
<image1>
Find the least possible answer to the subtraction shown.

Answer: 110


Question:
How many of the twelve pentominoes pictured below have at least one line of symmetry?
<image1>

Answer: 6

counting

Question:
How many of the twelve pentominoes pictured below have at least one line of symmetry?
<image1>

Answer: 6


Question:
In the (correct) calculation shown, some of the digits were replaced by the letters P, Q, R and S. What is the value of $P+Q+R+S$?
<image1>

Answer: 15

algebra

Question:
In the (correct) calculation shown, some of the digits were replaced by the letters P, Q, R and S. What is the value of $P+Q+R+S$?
<image1>

Answer: 15


Question:
A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order.  Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry.  Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it.  How many entries in the array are multiples of $ 67$?
<image1>

Answer: 17

algebra

Question:
A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order.  Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry.  Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it.  How many entries in the array are multiples of $ 67$?
<image1>

Answer: 17


Question:
Anna cuts the picture of a mushroom in two halves.
<image1>
She then arranges the two pieces together to form a new picture. What could this new picture look like?
<image2>

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

combinatorial geometry

Question:
Anna cuts the picture of a mushroom in two halves.
<image1>
She then arranges the two pieces together to form a new picture. What could this new picture look like?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Segment $AB$ measures 4 cm and is a diameter of circle $P$. In triangle $ABC$, point $C$ is on circle $P$ and $BC = 2$ cm. What is the area of the shaded region?

<image1>

Answer: 4\pi-2\sqrt{3}

Solution: The area of the shaded region equals the area of the circle minus the area of the triangle.  The area of the circle is $2^2\pi=4\pi$.  To find the area of the triangle, we look for information about the triangle.  Because angle $ACB$ intersects $180^\circ$ of the circle, we know $m\angle ACB=\frac{180^\circ}2=90^\circ$, so triangle $ACB$ is a right triangle.  What's more, since $AB=4$ and $BC=2$, it follows that $AC=2\sqrt{3}$ and the area of right triangle $ACB$ equals $\frac{2\cdot2\sqrt{3}}2=2\sqrt{3}$.  So, the area of the shaded region is $\boxed{4\pi - 2\sqrt{3}}$.

metric geometry - area

Question:
Segment $AB$ measures 4 cm and is a diameter of circle $P$. In triangle $ABC$, point $C$ is on circle $P$ and $BC = 2$ cm. What is the area of the shaded region?

<image1>

Answer: 4\pi-2\sqrt{3}


Question:
In the four squares of a row there always have to be exactly two coins. In the four squares below each other there also always have to be exactly two coins.
<image1>
On which square does one more coin have to be placed?

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

logic

Question:
In the four squares of a row there always have to be exactly two coins. In the four squares below each other there also always have to be exactly two coins.
<image1>
On which square does one more coin have to be placed?
(A) square $A$
(B) square $B$
(C) square $C$
(D) square $D$
(E) square $E$


Question:
How many different digits can you find in this picture?
<image1>

Answer: 6

counting

Question:
How many different digits can you find in this picture?
<image1>

Answer: 6


Question:
A $3 \times 3$ square initially has the number 0 in each of its cells. In one step all four numbers in one $2 \times 2$ sub-square such as the shaded one, for example, are then increased by 1. This operation is repeated several times to obtain the arrangement on the right. Unfortunately, some numbers in this arrangement are hidden. What number is in the square with the question mark?
<image1>

Answer: 16

logic

Question:
A $3 \times 3$ square initially has the number 0 in each of its cells. In one step all four numbers in one $2 \times 2$ sub-square such as the shaded one, for example, are then increased by 1. This operation is repeated several times to obtain the arrangement on the right. Unfortunately, some numbers in this arrangement are hidden. What number is in the square with the question mark?
<image1>

Answer: 16


Question:
In the diagram on the right, the triangle is equilateral.
<image1>
What is the area of the large circle divided by the area of the small circle?

Answer: 16

Solution: If $L$ is the centre of the large circle, $\angle J N L=30^{\circ}$ because $K N$ is a line of symmetry of the equilateral triangle. Since $L N$ and $L J$ are both radii, $\triangle J L N$ is isosceles and so $\angle N J L=\angle J N L=30^{\circ}$. Hence $\angle K L J=60^{\circ}$ and since $K L=J L$, $\triangle K J L$ is equilateral and $K M=M L$. Hence the radius of the larger circle is 4 times that of the smaller circle, and so the area is 16 times that of the smaller circle.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-024.jpg?height=499&width=508&top_left_y=2189&top_left_x=1248)

metric geometry - area

Question:
In the diagram on the right, the triangle is equilateral.
<image1>
What is the area of the large circle divided by the area of the small circle?

Answer: 16


Question: Find $AX$ in the diagram if $CX$ bisects $\angle ACB$. <image1>

Answer: 14

Solution: The Angle Bisector Theorem tells us that  \[\frac{AC}{AX}=\frac{BC}{BX}\]so \[AX=\frac{AC\cdot BX}{BC}=\frac{21\cdot30}{45}=\boxed{14}.\]

metric geometry - length

Question: Find $AX$ in the diagram if $CX$ bisects $\angle ACB$. <image1>

Answer: 14


Question:
A cuboid-shaped container that is not filled completely contains $120 \mathrm{~m}^{3}$ of water. The depth of the water is either $2 \mathrm{~m}$ or $3 \mathrm{~m}$ or $5 \mathrm{~m}$, depending on which side the container is actually standing on (drawings not to scale). How big is the volume of the container?
<image1>

Choices:
- $160 \mathrm{~m}^{3}$
- $180 \mathrm{~m}^{3}$
- $200 \mathrm{~m}^{3}$
- $220 \mathrm{~m}^{3}$
- $240 \mathrm{~m}^{3}$

solid geometry

Question:
A cuboid-shaped container that is not filled completely contains $120 \mathrm{~m}^{3}$ of water. The depth of the water is either $2 \mathrm{~m}$ or $3 \mathrm{~m}$ or $5 \mathrm{~m}$, depending on which side the container is actually standing on (drawings not to scale). How big is the volume of the container?
<image1>
(A) $160 \mathrm{~m}^{3}$
(B) $180 \mathrm{~m}^{3}$
(C) $200 \mathrm{~m}^{3}$
(D) $220 \mathrm{~m}^{3}$
(E) $240 \mathrm{~m}^{3}$


Question:
Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?
<image1>
<image2>

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

solid geometry

Question:
Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A village of 12 houses has four straight streets and four circular streets. The map shows 11 houses. In each straight street there are three houses and in each circular street there are also three houses. Where should the 12th house be placed on this map?
<image1>

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

logic

Question:
A village of 12 houses has four straight streets and four circular streets. The map shows 11 houses. In each straight street there are three houses and in each circular street there are also three houses. Where should the 12th house be placed on this map?
<image1>
(A) On A
(B) On B
(C) On C
(D) On D
(E) On E


Question:
Two chords $A B$ and $A C$ are drawn into a circle with diameter $\mathrm{AD}. \angle B A C=60^{\circ}$, $\overline{A B}=24 \mathrm{~cm}$, $\mathrm{E}$ lies on $\mathrm{AC}$ so that $\overline{E C}=3 \mathrm{~cm}$, and $\mathrm{BE}$ is perpendicular to $\mathrm{AC}$. How long is the chord $\mathrm{BD}$?
<image1>

Choices:
- $\sqrt{3} \mathrm{~cm}$
- $2 \mathrm{~cm}$
- $3 \mathrm{~cm}$
- $2 \sqrt{3} \mathrm{~cm}$
- $3 \sqrt{2} \mathrm{~cm}$

metric geometry - length

Question:
Two chords $A B$ and $A C$ are drawn into a circle with diameter $\mathrm{AD}. \angle B A C=60^{\circ}$, $\overline{A B}=24 \mathrm{~cm}$, $\mathrm{E}$ lies on $\mathrm{AC}$ so that $\overline{E C}=3 \mathrm{~cm}$, and $\mathrm{BE}$ is perpendicular to $\mathrm{AC}$. How long is the chord $\mathrm{BD}$?
<image1>
(A) $\sqrt{3} \mathrm{~cm}$
(B) $2 \mathrm{~cm}$
(C) $3 \mathrm{~cm}$
(D) $2 \sqrt{3} \mathrm{~cm}$
(E) $3 \sqrt{2} \mathrm{~cm}$


Question:
Freda's flying club designed a flag of a flying dove on a square grid as shown.
The area of the dove is $192 \mathrm{~cm}^{2}$. All parts of the perimeter of the dove are either quarter-circles or straight lines. What are the dimensions of the flag?
<image1>

Choices:
- $6 \mathrm{~cm} \times 4 \mathrm{~cm}$
- $12 \mathrm{~cm} \times 8 \mathrm{~cm}$
- $21 \mathrm{~cm} \times 14 \mathrm{~cm}$
- $24 \mathrm{~cm} \times 16 \mathrm{~cm}$
- $27 \mathrm{~cm} \times 18 \mathrm{~cm}$

Solution: Let each of the small squares in the grid have side-length $x \mathrm{~cm}$. Remove the shading and divide the dove into regions as shown.  It can be seen that the regions marked $\mathrm{A}$ and $\mathrm{B}$ combine to make a square of side $2 x \mathrm{~cm}$ and hence of area $4 x^{2} \mathrm{~cm}^{2}$. Similarly, regions C, D and E combine to make a rectangle with sides $2 x \mathrm{~cm}$ and $3 x \mathrm{~cm}$ and hence area $6 x^{2} \mathrm{~cm}^{2}$. Finally, region $\mathrm{F}$ is a rectangle with sides $2 x \mathrm{~cm}$ and $x \mathrm{~cm}$ and hence area $2 x^{2} \mathrm{~cm}^{2}$. Since the total area of the dove is $192 \mathrm{~cm}^{2}$, we have $4 x^{2}+6 x^{2}+2 x^{2}=192$ and hence $12 x^{2}=192$. Therefore $x^{2}=16$ and hence $x=4$. Hence the flag has length $(6 \times 4) \mathrm{cm}=24 \mathrm{~cm}$ and height $(4 \times 4) \mathrm{cm}=16 \mathrm{~cm}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-245.jpg?height=285&width=434&top_left_y=1565&top_left_x=1368)

combinatorial geometry

Question:
Freda's flying club designed a flag of a flying dove on a square grid as shown.
The area of the dove is $192 \mathrm{~cm}^{2}$. All parts of the perimeter of the dove are either quarter-circles or straight lines. What are the dimensions of the flag?
<image1>
(A) $6 \mathrm{~cm} \times 4 \mathrm{~cm}$
(B) $12 \mathrm{~cm} \times 8 \mathrm{~cm}$
(C) $21 \mathrm{~cm} \times 14 \mathrm{~cm}$
(D) $24 \mathrm{~cm} \times 16 \mathrm{~cm}$
(E) $27 \mathrm{~cm} \times 18 \mathrm{~cm}$


Question:
In a rectangle JKLM the angle bisector in $\mathrm{J}$ intersects the diagonal KM in $\mathrm{N}$. The distance of $\mathrm{N}$ to $\mathrm{LM}$ is 1 and the distance of $\mathrm{N}$ to $\mathrm{KL}$ is 8. How long is LM?
<image1>

Choices:
- $8+2 \sqrt{2}$
- $11-\sqrt{2}$
- 10
- $8+3 \sqrt{2}$
- $11+\frac{\sqrt{2}}{2}$

metric geometry - length

Question:
In a rectangle JKLM the angle bisector in $\mathrm{J}$ intersects the diagonal KM in $\mathrm{N}$. The distance of $\mathrm{N}$ to $\mathrm{LM}$ is 1 and the distance of $\mathrm{N}$ to $\mathrm{KL}$ is 8. How long is LM?
<image1>
(A) $8+2 \sqrt{2}$
(B) $11-\sqrt{2}$
(C) 10
(D) $8+3 \sqrt{2}$
(E) $11+\frac{\sqrt{2}}{2}$


Question:
A furniture shop sells 3-seater, 2-seater and 1-seater sofas that each have an equally wide armrest on the left and the right hand side. Each seat is equally wide (see picture). Together with the armrests the 3-seater sofa is $220 \mathrm{~cm}$ and the 2-seater sofa $160 \mathrm{~cm}$ wide. How wide is the 1-seater sofa?
<image1>

Choices:
- $60 \mathrm{~cm}$
- $80 \mathrm{~cm}$
- $90 \mathrm{~cm}$
- $100 \mathrm{~cm}$
- $120 \mathrm{~cm}$

solid geometry

Question:
A furniture shop sells 3-seater, 2-seater and 1-seater sofas that each have an equally wide armrest on the left and the right hand side. Each seat is equally wide (see picture). Together with the armrests the 3-seater sofa is $220 \mathrm{~cm}$ and the 2-seater sofa $160 \mathrm{~cm}$ wide. How wide is the 1-seater sofa?
<image1>
(A) $60 \mathrm{~cm}$
(B) $80 \mathrm{~cm}$
(C) $90 \mathrm{~cm}$
(D) $100 \mathrm{~cm}$
(E) $120 \mathrm{~cm}$


Question:
Belinda is building squares with matches adding small squares that it already has built according to the schema of the figure. How many matches does she have to add to the 30th square to build the 31st?
<image1>

Answer: 124

algebra

Question:
Belinda is building squares with matches adding small squares that it already has built according to the schema of the figure. How many matches does she have to add to the 30th square to build the 31st?
<image1>

Answer: 124


Question:
How many dots do all ladybirds have together?
<image1>

Answer: 19

counting

Question:
How many dots do all ladybirds have together?
<image1>

Answer: 19


Question: A triangular piece of paper is folded along the dotted line shown in the left-hand diagram to form the heptagon shown in the right-hand diagram. The total area of the shaded parts of the heptagon is $1 \mathrm{~cm}^{2}$. The area of the original triangle is $1 \frac{1}{2}$ times the area of the heptagon. What is the area of the original triangle, in $\mathrm{cm}^{2}$ ? <image1>

Answer: 3

Solution: Let $U$ be the unshaded area of the heptagon. Then the area of the triangle is $2 U+1$, as shown in the diagram. This is $1 \frac{1}{2}$ times the area of the heptagon, which is $U+1$, so we can form the equation $2 U+1=\frac{3}{2}(U+1)$. So $4 U+2=3 U+3$, hence $U=1$ and the area of the triangle is $2 \times 1+1=3$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-122.jpg?height=265&width=437&top_left_y=850&top_left_x=1369)

transformation geometry

Question: A triangular piece of paper is folded along the dotted line shown in the left-hand diagram to form the heptagon shown in the right-hand diagram. The total area of the shaded parts of the heptagon is $1 \mathrm{~cm}^{2}$. The area of the original triangle is $1 \frac{1}{2}$ times the area of the heptagon. What is the area of the original triangle, in $\mathrm{cm}^{2}$ ? <image1>

Answer: 3


Question:
Priti is learning a new language called Tedio. During her one hour lesson, which started at midday, she looks at the clock and notices that the hour hand and the minute hand make exactly the same angle with the vertical, as shown in the diagram. How many whole seconds remain until the end of the lesson?
<image1>

Answer: 276

Solution: Let the angle that each hand makes with the vertical be $x$ degrees and let the current time be $s$ seconds after midday. In one complete hour the hour hand will turn $30^{\circ}$. There are $60 \times 60=3600$ seconds in an hour so it takes $3600 \div 30=120$ seconds for the hour hand to turn one degree. Therefore $s=120 x$. In one complete hour the minute hand will turn $360^{\circ}$. There are 3600 seconds in an hour so it takes $3600 \div 360=10$ seconds for the minute hand to turn $1^{\circ}$. But the minute hand has turned clockwise through an angle of $(360-x)^{\circ}$. Therefore $s=10(360-x)$. Equating the two expressions we have obtained for $s$ we obtain the equation $120 x=10(360-x)$. The solution to this equation is $x=\frac{360}{13}$. Therefore the number of seconds elapsed since midday is $120 \times \frac{360}{13}=\frac{43200}{13}=3323 \frac{1}{13}$. The number of whole seconds remaining is $3600-3324=276$.

metric geometry - angle

Question:
Priti is learning a new language called Tedio. During her one hour lesson, which started at midday, she looks at the clock and notices that the hour hand and the minute hand make exactly the same angle with the vertical, as shown in the diagram. How many whole seconds remain until the end of the lesson?
<image1>

Answer: 276


Question:
Each of the points $A$, $B$, $C$, $D$, $E$, and $F$ in the figure below represent a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along the line each are added to produce 5 sums, one for each line. The total of the sums is $47$. What is the digit represented by $B$?

<image1>

Answer: 5

algebra

Question:
Each of the points $A$, $B$, $C$, $D$, $E$, and $F$ in the figure below represent a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along the line each are added to produce 5 sums, one for each line. The total of the sums is $47$. What is the digit represented by $B$?

<image1>

Answer: 5


Question:
Daniela has got cubes with their edges $1 \mathrm{dm}$ long. She has put some of them into the aquarium of the shape of a cube with the edges $3 \mathrm{dm}$ long as you see in the picture. How much more cubes can she put into the aquarium?
<image1>

Answer: 17

solid geometry

Question:
Daniela has got cubes with their edges $1 \mathrm{dm}$ long. She has put some of them into the aquarium of the shape of a cube with the edges $3 \mathrm{dm}$ long as you see in the picture. How much more cubes can she put into the aquarium?
<image1>

Answer: 17


Question:
John made a construction with wooden cubes of the same size, with the three views shown beside, using as many cubes as possible. Ana, John's sister, wants to remove all the cubes she can, without modifying these three views. At most, how many cubes can she remove?
front:
<image1>
right side:
<image2>
above:
<image3>

Answer: 12

descriptive geometry

Question:
John made a construction with wooden cubes of the same size, with the three views shown beside, using as many cubes as possible. Ana, John's sister, wants to remove all the cubes she can, without modifying these three views. At most, how many cubes can she remove?
front:
<image1>
right side:
<image2>
above:
<image3>

Answer: 12


Question: When the 5 pieces <image1> are fitted together correctly, the result is a rectangle with a calculation written on it. What is the answer to this calculation?

Answer: -100

arithmetic

Question: When the 5 pieces <image1> are fitted together correctly, the result is a rectangle with a calculation written on it. What is the answer to this calculation?

Answer: -100


Question:
A butterfly sat down on a correctly solved exercise. What number is the butterfly covering?
<image1>

Answer: 500

arithmetic

Question:
A butterfly sat down on a correctly solved exercise. What number is the butterfly covering?
<image1>

Answer: 500


Question:
The curved surfaces of two identical cylinders are cut open along the vertical dotted line, as shown and then stuck together to create the curved surface of one big cylinder. What can be said about the volume of the resulting cylinder compared to the volume of one of the small cylinders?
<image1>

Choices:
- It is 2-times as big.
- It is 3-times as big.
- It is $\pi$-times as big.
- It is 4-times as big.
- It is 8-times as big.

solid geometry

Question:
The curved surfaces of two identical cylinders are cut open along the vertical dotted line, as shown and then stuck together to create the curved surface of one big cylinder. What can be said about the volume of the resulting cylinder compared to the volume of one of the small cylinders?
<image1>
(A) It is 2-times as big.
(B) It is 3-times as big.
(C) It is $\pi$-times as big.
(D) It is 4-times as big.
(E) It is 8-times as big.


Question:
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
<image1>

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

metric geometry - length

Question:
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
<image1>
(A) $2\sqrt{5} - \sqrt{3}$
(B) $3$
(C) $\frac{3\sqrt{7} - \sqrt{3}}{2}$
(D) $2\sqrt{3}$
(E) $\frac{5 + 2\sqrt{3}}{2}$


Question:
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively.
<image1>
Then the measure of $AB$ is equal to

Choices:
- $\frac{1}{2}a+2b$
- $\frac{3}{2}b+\frac{3}{4}a$
- $2a-b$
- $4b-\frac{1}{2}a$
- $a+b$

metric geometry - angle

Question:
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively.
<image1>
Then the measure of $AB$ is equal to
(A) $\frac{1}{2}a+2b$
(B) $\frac{3}{2}b+\frac{3}{4}a$
(C) $2a-b$
(D) $4b-\frac{1}{2}a$
(E) $a+b$


Question: The shown triangle $A B C$ is isosceles with $\measuredangle A B C=40^{\circ}$. The two angles indicated $\measuredangle E A B$ and $\measuredangle D C A$ are equally big. How big is the angle $\measuredangle C F E$ ? <image1>

Choices:
- $55^{\circ}$
- $60^{\circ}$
- $65^{\circ}$
- $70^{\circ}$
- $75^{\circ}$

metric geometry - angle

Question:
The shown triangle $A B C$ is isosceles with $\measuredangle A B C=40^{\circ}$. The two angles indicated $\measuredangle E A B$ and $\measuredangle D C A$ are equally big. How big is the angle $\measuredangle C F E$ ? <image1>
(A) $55^{\circ}$
(B) $60^{\circ}$
(C) $65^{\circ}$
(D) $70^{\circ}$
(E) $75^{\circ}$


Question:
Using the pieces $A, B, C, D$ and $E$ one can fill this shape completely: Which of the pieces lies on the dot?
<image1>
<image2>

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

combinatorial geometry

Question:
Using the pieces $A, B, C, D$ and $E$ one can fill this shape completely: Which of the pieces lies on the dot?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\fracmn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

<image1>

Answer: 109

metric geometry - area

Question:
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\fracmn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

<image1>

Answer: 109


Question:
Which two building blocks can be joined together so that the object shown is created?
<image1>
<image2>

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

combinatorial geometry

Question:
Which two building blocks can be joined together so that the object shown is created?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square, The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$?

<image1>

Answer: 68

metric geometry - area

Question:
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square, The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$?

<image1>

Answer: 68


Question:
Into how many pieces will the string be cut?
<image1>

Answer: 9

counting

Question:
Into how many pieces will the string be cut?
<image1>

Answer: 9


Question:
The following picture shows a necklace with six pearls:
<image1>
Which of the following diagrams shows the same necklace?
<image2>

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

topology

Question:
The following picture shows a necklace with six pearls:
<image1>
Which of the following diagrams shows the same necklace?
<image2>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
In the diagram we see 10 islands that are connected by 15 bridges. What is the minimum number of bridges that need to be closed off so that there is no connection from $A$ to $B$ anymore?
<image1>

Answer: 3

graph theory

Question:
In the diagram we see 10 islands that are connected by 15 bridges. What is the minimum number of bridges that need to be closed off so that there is no connection from $A$ to $B$ anymore?
<image1>

Answer: 3


Question:
There are 10 balls, numbered 0 to 9 in a basket. John and George play a game. Each person is allowed to take three balls from the basket and calculate the total of the numbers on the balls. What is the biggest possible difference between the john and Georges totals?
<image1>

Answer: 21

algebra

Question:
There are 10 balls, numbered 0 to 9 in a basket. John and George play a game. Each person is allowed to take three balls from the basket and calculate the total of the numbers on the balls. What is the biggest possible difference between the john and Georges totals?
<image1>

Answer: 21


Question: A cube is being rolled on a plane so it turns around its edges. Its bottom face passes through the positions $1,2,3,4,5,6$ and 7 in that order, as shown. Which of these two positions were occupied by the same face of the cube? <image1>

Choices:
- 1 and 7
- 1 and 6
- 1 and 5
- 2 and 7
- 2 and 6

Solution: Imagine the grid is sticky so that when the cube rolls over it, each cell of the grid fastens to the face of the cube touching it. The result would be equivalent to taking the arrangement of cells as shown, cutting it out and folding it into a cube. The latter is possible (for example) with 5 on  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-147.jpg?height=214&width=362&top_left_y=1606&top_left_x=1418) the bottom, 6 at the back, 7 on the right, 4 on the left, 3 at the front, 2 on the top and 1 folding over the 6 at the back. Hence 1 and 6 are occupied by the same face of the cube.

solid geometry

Question:
A cube is being rolled on a plane so it turns around its edges. Its bottom face passes through the positions $1,2,3,4,5,6$ and 7 in that order, as shown. Which of these two positions were occupied by the same face of the cube? <image1>
(A) 1 and 7
(B) 1 and 6
(C) 1 and 5
(D) 2 and 7
(E) 2 and 6


Question:
In the diagram one can see my decision-die in three different positions. What is the probability I get a "YES", when rolling this die once.
<image1>

Choices:
- $\frac{1}{3}$
- $\frac{1}{2}$
- $\frac{5}{9}$
- $\frac{2}{3}$
- $\frac{5}{6}$

solid geometry

Question:
In the diagram one can see my decision-die in three different positions. What is the probability I get a "YES", when rolling this die once.
<image1>
(A) $\frac{1}{3}$
(B) $\frac{1}{2}$
(C) $\frac{5}{9}$
(D) $\frac{2}{3}$
(E) $\frac{5}{6}$


Question:
Triangle $A B C$ is equilateral and has area 9. The dividing lines are parallel to the sides, and divide the sides into three equal lengths. What is the area of the grey shaded part of the triangle?
<image1>

Answer: 6

combinatorial geometry

Question:
Triangle $A B C$ is equilateral and has area 9. The dividing lines are parallel to the sides, and divide the sides into three equal lengths. What is the area of the grey shaded part of the triangle?
<image1>

Answer: 6


Question:
<image1>

Which cylinder has twice the volume of the cylinder shown above?

<image2>

Choices:
- A
- B
- C
- D
- None of the above

solid geometry

Question:
<image1>

Which cylinder has twice the volume of the cylinder shown above?

<image2>
(A) A
(B) B
(C) C
(D) D
(E) None of the above


Question: Consider a $1$ by $2$ by $3$ rectangular prism. Find the length of the shortest path between opposite corners $A$ and $B$ that does not leave the surface of the prism.\n<image1>

Answer: $3\sqrt{2}$

solid geometry

Question: Consider a $1$ by $2$ by $3$ rectangular prism. Find the length of the shortest path between opposite corners $A$ and $B$ that does not leave the surface of the prism.\n<image1>

Answer: $3\sqrt{2}$


Question:
The 8 corners of the shape in the picture are to be labelled with the numbers 1, 2, 3 or 4 , so that the numbers at the ends of each of the lines shown are different. How often does the number 4 appear on the shape?
<image1>

Answer: 4

combinatorics

Question:
The 8 corners of the shape in the picture are to be labelled with the numbers 1, 2, 3 or 4 , so that the numbers at the ends of each of the lines shown are different. How often does the number 4 appear on the shape?
<image1>

Answer: 4


Question:
Each participant in a cooking contest baked one tray of cookies like the one shown beside. What is the smallest number of trays of cookies needed to make the following plate?
<image1>

Answer: 3

counting

Question:
Each participant in a cooking contest baked one tray of cookies like the one shown beside. What is the smallest number of trays of cookies needed to make the following plate?
<image1>

Answer: 3


Question:
The picture shows the clown Dave dancing on top of two balls and one cubic box. The radius of the lower ball is $6 \mathrm{dm}$, the radius of the upper ball is three times less. The side of the cubic box is $4 \mathrm{dm}$ longer than the radius of the upper ball. At what height (in $\mathrm{dm}$ ) above the ground is the clown Dave standing?
<image1>

Answer: 22

algebra

Question:
The picture shows the clown Dave dancing on top of two balls and one cubic box. The radius of the lower ball is $6 \mathrm{dm}$, the radius of the upper ball is three times less. The side of the cubic box is $4 \mathrm{dm}$ longer than the radius of the upper ball. At what height (in $\mathrm{dm}$ ) above the ground is the clown Dave standing?
<image1>

Answer: 22


Question: Two squares of the same size, and with their edges parallel, cover a circle with a radius of $3 \mathrm{~cm}$, as shown. In square centimetres, what is the total shaded area? <image1>

Choices:
- $8(\pi-1)$
- $6(2 \pi-1)$
- $(9 \pi-25)$
- $9(\pi-2)$
- $\frac{6\pi}{5}$

Solution: From the diagram, it can be seen that the area of the central square is half of the dashed square, that is $\frac{1}{2}\times 6\times 6=18$. The shaded area is the area of the circle less the area of the central square, so is $\pi\times 3^2-18=9(\pi -2)$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-010.jpg?height=410&width=490&top_left_y=520&top_left_x=1320)

metric geometry - area

Question:
Two squares of the same size, and with their edges parallel, cover a circle with a radius of $3 \mathrm{~cm}$, as shown. In square centimetres, what is the total shaded area? <image1>
(A) $8(\pi-1)$
(B) $6(2 \pi-1)$
(C) $(9 \pi-25)$
(D) $9(\pi-2)$
(E) $\frac{6\pi}{5}$


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

Answer: 9

combinatorics

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

Answer: 9


Question:
The diagram shows two circles and a square with sides of length $10 \mathrm{~cm}$. One vertex of the square is at the centre of the large circle and two sides of the square are tangents to both circles. The small circle touches the large circle. The radius of the small circle is $(a-b \sqrt{2}) \mathrm{cm}$.
<image1>
What is the value of $a+b$ ?

Answer: 50

Solution: Let $O$ and $P$ be the centres of the large and small circles respectively and label points $Q$ and $S$ as shown in the diagram. Let the radius of the small circle be $r \mathrm{~cm}$. Draw line $P R$ so that $R$ is on $Q S$ and $P R$ is parallel to $O S$. Draw in line $O Q$. Since triangle $O Q S$ is right-angled and isosceles, $O Q^{2}=10^{2}+10^{2}$ by Pythagoras. Hence $O Q=10 \sqrt{2} \mathrm{~cm}$. Similarly, since triangle $P Q R$ is right- ![](https://cdn.mathpix.com/cropped/2023_12_27_d884587740e101721faeg-2.jpg?height=311&width=325&top_left_y=193&top_left_x=1525) angled and isosceles, $P Q=r \sqrt{2} \mathrm{~cm}$. Note that angle $O Q S=$ angle $P Q S=45^{\circ}$ so $O P Q$ is a straight line. Therefore $10 \sqrt{2}=10+r+r \sqrt{2}$. This has solution $$ r=\frac{10(\sqrt{2}-1)}{\sqrt{2}+1}=\frac{10(\sqrt{2}-1)(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}=\frac{10(2+1-2 \sqrt{2})}{2-1}=30-20 \sqrt{2} $$ Hence the radius of the small circle is $(30-20 \sqrt{2}) \mathrm{cm}$ and the value of $a+b$ is $30+20=50$.

metric geometry - length

Question:
The diagram shows two circles and a square with sides of length $10 \mathrm{~cm}$. One vertex of the square is at the centre of the large circle and two sides of the square are tangents to both circles. The small circle touches the large circle. The radius of the small circle is $(a-b \sqrt{2}) \mathrm{cm}$.
<image1>
What is the value of $a+b$ ?

Answer: 50


Question:
Harry folds a sheet of paper five times. Then he makes a hole in the folded paper, after which he unfolds it.
<image1>
How many holes has the unfolded paper?

Answer: 32

arithmetic

Question:
Harry folds a sheet of paper five times. Then he makes a hole in the folded paper, after which he unfolds it.
<image1>
How many holes has the unfolded paper?

Answer: 32