Question:
The squares are formed by intersecting the segment $A B$ of $24 \mathrm{~cm}$ by the broken line $A A_{1} A_{2} \ldots A_{12} B$ (see the figure). Find the length of $A A_{1} A_{2} \ldots A_{12} B$.
<image1>

Choices:
- $48 \mathrm{~cm}$
- $72 \mathrm{~cm}$
- $96 \mathrm{~cm}$
- $56 \mathrm{~cm}$
- $106 \mathrm{~cm}$

metric geometry - length

Question:
The squares are formed by intersecting the segment $A B$ of $24 \mathrm{~cm}$ by the broken line $A A_{1} A_{2} \ldots A_{12} B$ (see the figure). Find the length of $A A_{1} A_{2} \ldots A_{12} B$.
<image1>
(A) $48 \mathrm{~cm}$
(B) $72 \mathrm{~cm}$
(C) $96 \mathrm{~cm}$
(D) $56 \mathrm{~cm}$
(E) $106 \mathrm{~cm}$


Question: Two chords $P Q$ and $P R$ are drawn in a circle with diameter $P S$. The point $T$ lies on $P R$ and $Q T$ is perpendicular to $P R$. The angle $Q P R=60^{\circ}, P Q=24 \mathrm{~cm}, R T=3 \mathrm{~cm}$. What is the length of the chord $Q S$ in $\mathrm{cm}$ ? <image1>

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

Solution: $P S$ is a diameter, so $\angle P R S=90^{\circ}$ (angle in a semicircle). Let $U$ be the point on $Q T$ for which $S U$ is perpendicular to $Q T$. Hence RSUT is a rectangle, and $S U=T R=3$.  In triangle $T P Q$, we have $\angle P T Q=90^{\circ}$ and $\angle Q P T=60^{\circ}$, so $\angle T Q P=30^{\circ}$. Also, $\angle P Q S=90^{\circ}$ (angle in a semicircle) so $\angle U Q S=90^{\circ}-\angle T Q P=90^{\circ}-30^{\circ}=60^{\circ}$. Also $\sin \angle U Q S=\frac{U S}{Q S}=\frac{3}{Q S}$. So  $$ Q S=\frac{3}{\sin \angle U Q S}=\frac{3}{\frac{1}{2} \sqrt{3}}=3 \times \frac{2}{\sqrt{3}}=2 \sqrt{3} $$  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-254.jpg?height=503&width=520&top_left_y=1956&top_left_x=1359)

metric geometry - length

Question:
Two chords $P Q$ and $P R$ are drawn in a circle with diameter $P S$. The point $T$ lies on $P R$ and $Q T$ is perpendicular to $P R$. The angle $Q P R=60^{\circ}, P Q=24 \mathrm{~cm}, R T=3 \mathrm{~cm}$. What is the length of the chord $Q S$ in $\mathrm{cm}$ ? <image1>
(A) $\sqrt{3}$
(B) 2
(C) 3
(D) $2 \sqrt{3}$
(E) $3 \sqrt{2}$


Question:
Martin has three cards that are labelled on both sides with a number. Martin places the three cards on the table without paying attention to back or front. He adds the three numbers that he can then see. How many different sums can Martin get that way?
<image1>

Choices:
- 3
- 5
- 6
- 9
- A different amount.

combinatorics

Question:
Martin has three cards that are labelled on both sides with a number. Martin places the three cards on the table without paying attention to back or front. He adds the three numbers that he can then see. How many different sums can Martin get that way?
<image1>
(A) 3
(B) 5
(C) 6
(D) 9
(E) A different amount.


Question:
Alice subtracts one two-digit number from another two-digit number. Afterwards she paints over two digits in the calculation. How big is the sum of the two painted digits?
<image1>

Answer: 13

algebra

Question:
Alice subtracts one two-digit number from another two-digit number. Afterwards she paints over two digits in the calculation. How big is the sum of the two painted digits?
<image1>

Answer: 13


Question:
The circles of the figure should be numbered from 0 to 10 , each with a different number. The five sums of the three numbers written on each diameter must be odd numbers. If one of these sums is the smallest possible, what will be the largest possible value of one of the remaining sums?
<image1>

Answer: 21

algebra

Question:
The circles of the figure should be numbered from 0 to 10 , each with a different number. The five sums of the three numbers written on each diameter must be odd numbers. If one of these sums is the smallest possible, what will be the largest possible value of one of the remaining sums?
<image1>

Answer: 21


Question:
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated.  Which city had the greatest percentage increase in population from 1970 to 1980?

<image1>

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

statistics

Question:
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated.  Which city had the greatest percentage increase in population from 1970 to 1980?

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


Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. How many subsets of these seven edges form a connected figure?\n<image1>

Answer: 81

Solution: We break this into cases. First, if the middle edge is not included, then there are 6 ∗ 5 = 30 ways to choose two distinct points for the figure to begin and end at. We could also allow the figure to include all or none of the six remaining edges, for a total of 32 connected figures not including the middle edge. Now let's assume we are including the middle edge. Of the three edges to the left of the middle edge, there are 7 possible subsets we can include (8 total subsets, but we subtract off the subset consisting of only the edge parallel to the middle edge since it's not connected). Similarly, of the three edges to the right of the middle edge, there are 7 possible subsets we can include. In total, there are 49 possible connected figures that include the middle edge. Therefore, there are 32 + 49 = 81 possible connected figures.

combinatorial geometry

Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. How many subsets of these seven edges form a connected figure?\n<image1>

Answer: 81


Question:
When simon the squirrel comes down from his tree onto the floor, he never moves further than $5 \mathrm{~m}$ away from the trunk of his tree. Furthermore, he stays at least $5 \mathrm{~m}$ away from the dog kennel. Which picture shows most accurately the area in which Simon can be found?
<image1>

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

metric geometry - area

Question:
When simon the squirrel comes down from his tree onto the floor, he never moves further than $5 \mathrm{~m}$ away from the trunk of his tree. Furthermore, he stays at least $5 \mathrm{~m}$ away from the dog kennel. Which picture shows most accurately the area in which Simon can be found?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Assume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of Lena, Wisconsin, is at $45^{\circ}$ North Latitude, exactly halfway between the equator and the North Pole. What is the number of miles in the circumference of the circle on Earth parallel to the equator and through Lena, Wisconsin? Express your answer to the nearest hundred miles.  (You may use a calculator for this problem.)

<image1>

Answer: 17700

Solution:
Let Earth's radius be $r$.  Since the equator measures 25100 miles, we have $2\pi r = 25100 \Rightarrow r = \frac{12550}{\pi}$.

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Let Earth's center be $A$, let the center of the circle that passes through Lena be $B$, and let Lena be $L$.  Because $\overline{BL}$ is parallel to the equator and Lena is at $45^\circ$ North Latitude, $\triangle ABL$ is a 45-45-90 triangle.  Thus, $BL=AB=\frac{r}{\sqrt{2}}$.

The number of miles in the circumference of the circle parallel to the equator and through Lena is $2\pi \cdot BL = 2\pi \frac{r}{\sqrt{2}} = \frac{25100}{\sqrt{2}} \approx 17748$ miles.  To the nearest hundred miles, this value is $\boxed{17700}$ miles.

metric geometry - length

Question:
Assume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of Lena, Wisconsin, is at $45^{\circ}$ North Latitude, exactly halfway between the equator and the North Pole. What is the number of miles in the circumference of the circle on Earth parallel to the equator and through Lena, Wisconsin? Express your answer to the nearest hundred miles.  (You may use a calculator for this problem.)

<image1>

Answer: 17700


Question:
On an idealised rectangular billiard table with side lengths $3 \mathrm{~m}$ and $2 \mathrm{m}$ a ball (point-shaped) is pushed away from point $M$ on the long side $A B$. It is reflected exactly once on each of the other sides as shown. at which distance from the vertex $A$ will the ball hit this side again if $B M=1.2 \mathrm{~m}$ and $B N=$ $0.8 m$?
<image1>

Choices:
- $2 \mathrm{~m}$
- $1.5 \mathrm{~m}$
- $1.2 \mathrm{~m}$
- $2.8 \mathrm{~m}$
- $1.8 \mathrm{~m}$

metric geometry - length

Question:
On an idealised rectangular billiard table with side lengths $3 \mathrm{~m}$ and $2 \mathrm{m}$ a ball (point-shaped) is pushed away from point $M$ on the long side $A B$. It is reflected exactly once on each of the other sides as shown. at which distance from the vertex $A$ will the ball hit this side again if $B M=1.2 \mathrm{~m}$ and $B N=$ $0.8 m$?
<image1>
(A) $2 \mathrm{~m}$
(B) $1.5 \mathrm{~m}$
(C) $1.2 \mathrm{~m}$
(D) $2.8 \mathrm{~m}$
(E) $1.8 \mathrm{~m}$


Question: Alice has four jigsaw pieces. <image1> Which two can be fitted together to form a hexagon?

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

combinatorial geometry

Question:
Alice has four jigsaw pieces. <image1> Which two can be fitted together to form a hexagon?
(A) 1 and 2
(B) 1 and 3
(C) 2 and 3
(D) 2 and 4
(E) 1 and 4


Question:
On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

<image1>

Choices:
- $\frac{17}{36}$
- $\frac{35}{72}$
- $\frac{1}{2}$
- $\frac{37}{72}$
- $\frac{19}{36}$

combinatorics

Question:
On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

<image1>
(A) $\frac{17}{36}$
(B) $\frac{35}{72}$
(C) $\frac{1}{2}$
(D) $\frac{37}{72}$
(E) $\frac{19}{36}$


Question:
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).
<image1>

The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?

Answer: 337

analytic geometry

Question:
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).
<image1>

The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?

Answer: 337


Question:
The diagram consists of three circles of equal radius $R$. The centre of those circles lie on a common straight line where the middle circle goes through the centres of the other two circles (see diagram). How big is the perimeter of the figure?
<image1>

Choices:
- $\frac{10 \pi R}{3}$
- $\frac{5 \pi R}{3}$
- $\frac{2 \pi R \sqrt{3}}{3}$
- $2 \pi R \sqrt{3}$
- $4 \pi R$

metric geometry - length

Question:
The diagram consists of three circles of equal radius $R$. The centre of those circles lie on a common straight line where the middle circle goes through the centres of the other two circles (see diagram). How big is the perimeter of the figure?
<image1>
(A) $\frac{10 \pi R}{3}$
(B) $\frac{5 \pi R}{3}$
(C) $\frac{2 \pi R \sqrt{3}}{3}$
(D) $2 \pi R \sqrt{3}$
(E) $4 \pi R$


Question:
Mike has built a construction, shown in the upper picture, from equal cubes. Lily has taken several cubes out of it, thus Mike's construction became such as we see in the lower picture. How many cubes has Lily taken?
<image1>

Answer: 7

solid geometry

Question:
Mike has built a construction, shown in the upper picture, from equal cubes. Lily has taken several cubes out of it, thus Mike's construction became such as we see in the lower picture. How many cubes has Lily taken?
<image1>

Answer: 7


Question:
One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling's car and her house over the course of her trip?

<image1>

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

statistics

Question:
One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling's car and her house over the course of her trip?

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


Question:
$\angle 1 + \angle 2 = 180^\circ $

$\angle 3 = \angle 4$

Find $\angle 4$.

<image1>

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

metric geometry - angle

Question:
$\angle 1 + \angle 2 = 180^\circ $

$\angle 3 = \angle 4$

Find $\angle 4$.

<image1>
(A) $20^\circ$
(B) $25^\circ$
(C) $30^\circ$
(D) $35^\circ$
(E) $40^\circ$


Question:
Mrs. Maisl buys four pieces of corn-on-the-cob for each of the four members of her family and get the discount offered. How much does she end up paying?
<image1>

Choices:
- $0.80 €$
- $1.20 €$
- $2.80 €$
- $3.20 €$
- $80 €$

arithmetic

Question:
Mrs. Maisl buys four pieces of corn-on-the-cob for each of the four members of her family and get the discount offered. How much does she end up paying?
<image1>
(A) $0.80 €$
(B) $1.20 €$
(C) $2.80 €$
(D) $3.20 €$
(E) $80 €$


Question: Let $T$ be $7$. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$, find $k$.\n<image1>

Answer: 12

Solution: $\frac{\pi \cdot 7^2 - \pi}{4} = 12 \pi$\n$k = 12$

metric geometry - area

Question: Let $T$ be $7$. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$, find $k$.\n<image1>

Answer: 12


Question:
Seven identical dice (each with 1, 2, 3, 4, 5 and 6 points on their faces) are glued together to form the solid shown. Faces that are glued together each have the same number of points. How many points can be seen on the surface of the solid?
<image1>

Answer: 105

solid geometry

Question:
Seven identical dice (each with 1, 2, 3, 4, 5 and 6 points on their faces) are glued together to form the solid shown. Faces that are glued together each have the same number of points. How many points can be seen on the surface of the solid?
<image1>

Answer: 105


Question:
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$
<image1>

Answer: 9

metric geometry - angle

Question:
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$
<image1>

Answer: 9


Question:
A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?

<image1>

Answer: 353

combinatorial geometry

Question:
A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?

<image1>

Answer: 353


Question:
The diagram shows a parallelogram $W X Y Z$ with area $S$. The diagonals of the parallelogram meet at the point $O$. The point $M$ is on the edge $Z Y$. The lines $W M$ and $Z X$ meet at $N$. The lines $M X$ and $W Y$ meet at $P$. The sum of the areas of triangles $W N Z$ and $X Y P$ is $\frac{1}{3} S$. What is the area of quadrilateral MNOP ?
<image1>

Choices:
- $\frac{1}{6} S$
- $\frac{1}{8} S$
- $\frac{1}{10} S$
- $\frac{1}{12} S$
- $\frac{1}{14} S$

Solution: The area of parallelogram $W X Y Z$ is $S$. Therefore the area of triangle $W X M$, which has the same base and height, is $\frac{1}{2} S$. Hence the sum of the areas of triangle $W M Z$ and triangle $X Y M$ is also $\frac{1}{2} S$. The sum of the areas of triangle $W N Z$ and triangle $X Y P$ is given as $\frac{1}{3} S$ and therefore the sum of the areas of triangle $Z N M$ and triangle $M P Y$ is $\frac{1}{2} S-\frac{1}{3} S=\frac{1}{6} S$. The area of triangle $Z O Y$, which has the same base as the parallelogram but only half the height is $\frac{1}{2} \times \frac{1}{2} S=\frac{1}{4} S$. Therefore the area of quadrilateral MNOP is $\frac{1}{4} S-\frac{1}{6} S=\frac{1}{12} S$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-232.jpg?height=362&width=1238&top_left_y=236&top_left_x=364)

metric geometry - area

Question:
The diagram shows a parallelogram $W X Y Z$ with area $S$. The diagonals of the parallelogram meet at the point $O$. The point $M$ is on the edge $Z Y$. The lines $W M$ and $Z X$ meet at $N$. The lines $M X$ and $W Y$ meet at $P$. The sum of the areas of triangles $W N Z$ and $X Y P$ is $\frac{1}{3} S$. What is the area of quadrilateral MNOP ?
<image1>
(A) $\frac{1}{6} S$
(B) $\frac{1}{8} S$
(C) $\frac{1}{10} S$
(D) $\frac{1}{12} S$
(E) $\frac{1}{14} S$


Question:
Which of the "buildings" A-E, each consisting of 5 cubes, cannot be obtained from the building on the right, if you are allowed to move only one cube?
<image1>
<image2>

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

descriptive geometry

Question:
Which of the "buildings" A-E, each consisting of 5 cubes, cannot be obtained from the building on the right, if you are allowed to move only one cube?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The figure shows a circle with the diameter $A B$ and point $D$ on it. Find $d$.
<image1>

Answer: 4

analytic geometry

Question:
The figure shows a circle with the diameter $A B$ and point $D$ on it. Find $d$.
<image1>

Answer: 4


Question:
The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
<image1>

<image2>

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

analytic geometry

Question:
The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
<image1>

<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The diagram below shows five rectangles, each containing some of the letters $\mathrm{P}, \mathrm{R}, \mathrm{I}, \mathrm{S}$ and $\mathrm{M}$.
<image1>
Harry wants to cross out letters so that each rectangle contains only one letter and each rectangle contains a different letter. Which letter does he not cross out in rectangle 2?

Choices:
- P
- R
- I
- S
- M

Solution: Rectangle 4 only contains one letter and hence letter $\mathrm{S}$ must be crossed out in any other rectangle. Therefore letter $\mathrm{P}$ is the only letter left in rectangle 1 and must be crossed out in all the other rectangles. This means letter I is the only one left in rectangle 3 and must be crossed out in all other rectangles. This leaves letter R not crossed out in rectangle 2.

logic

Question:
The diagram below shows five rectangles, each containing some of the letters $\mathrm{P}, \mathrm{R}, \mathrm{I}, \mathrm{S}$ and $\mathrm{M}$.
<image1>
Harry wants to cross out letters so that each rectangle contains only one letter and each rectangle contains a different letter. Which letter does he not cross out in rectangle 2?
(A) P
(B) R
(C) I
(D) S
(E) M


Question:
How many different ways can you follow from point $A$ to point $B$ if you you can go only down, right or down diagonally by the sides of small triangles?
<image1>

Answer: 90

combinatorics

Question:
How many different ways can you follow from point $A$ to point $B$ if you you can go only down, right or down diagonally by the sides of small triangles?
<image1>

Answer: 90


Question:
Else has two machines R and S. If she puts a square piece of paper into machine $R$ it is rotated: <image1>
If she puts the piece of paper in machine $S$ it is printed on: <image2>
She wants to produce the following picture: <image3>
In which order does Else use the two machines so that she gets this picture?
<image4>

Choices:
- SRR
- RSR
- RSS
- RRS
- SRS

transformation geometry

Question:
Else has two machines R and S. If she puts a square piece of paper into machine $R$ it is rotated: <image1>
If she puts the piece of paper in machine $S$ it is printed on: <image2>
She wants to produce the following picture: <image3>
In which order does Else use the two machines so that she gets this picture?
<image4>
(A) SRR
(B) RSR
(C) RSS
(D) RRS
(E) SRS


Question:
The figure may be folded along the lines shown to form a number cube.  Three number faces come together at each corner of the cube.  What is the largest sum of three numbers whose faces come together at a corner?

<image1>

Answer: 14

solid geometry

Question:
The figure may be folded along the lines shown to form a number cube.  Three number faces come together at each corner of the cube.  What is the largest sum of three numbers whose faces come together at a corner?

<image1>

Answer: 14


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

Answer: 9

counting

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

Answer: 9


Question:
The point $O$ is the center of the circle in the picture. What is the diameter of the circle?
<image1>

Answer: 10

metric geometry - length

Question:
The point $O$ is the center of the circle in the picture. What is the diameter of the circle?
<image1>

Answer: 10


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

Answer: 6

combinatorics

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

Answer: 6


Question:
Rose the cat walks along the wall. She starts at point $B$ and follows the direction of the arrows shown in the picture. The cat walks a total of 20 metres. Where does she end up?
<image1>

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

metric geometry - length

Question:
Rose the cat walks along the wall. She starts at point $B$ and follows the direction of the arrows shown in the picture. The cat walks a total of 20 metres. Where does she end up?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The picture shows a rectangle with four identical triangles. Determine the total area of the triangles.
<image1>

Choices:
- $46 \mathrm{~cm}^{2}$
- $52 \mathrm{~cm}^{2}$
- $54 \mathrm{~cm}^{2}$
- $56 \mathrm{~cm}^{2}$
- $64 \mathrm{~cm}^{2}$

metric geometry - area

Question:
The picture shows a rectangle with four identical triangles. Determine the total area of the triangles.
<image1>
(A) $46 \mathrm{~cm}^{2}$
(B) $52 \mathrm{~cm}^{2}$
(C) $54 \mathrm{~cm}^{2}$
(D) $56 \mathrm{~cm}^{2}$
(E) $64 \mathrm{~cm}^{2}$


Question:
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square?

<image1>

Answer: 117

metric geometry - area

Question:
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square?

<image1>

Answer: 117


Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.\n<image1>

Answer: $\boxed{\frac{3}{4}}$

Solution: So trivially if $P$ is in the square on the right, then the line will go through both squares. Now consider the square on the left.\nThe way in which $P$ won't satisfy the condition is if the line goes through the left square and then leaves before passing through square $2$\nWe can rewrite this numerically as if some line $l$ intersects the line $y=1$ at some point $(x_0,1)$ such that $0\le{x_0}\le{1}$.\nAlso we can represent $l$ as $y=\frac{x}{2}+b$. Now we can solve for $b$ in terms of $x$ by plugging in $y=1$ and rearranging.\nThis gives $2-2b=x$. Thus we want $\frac{1}{2}<b<1$. Now we want to compute the total interval $b$ can take on. By using the extreme points of $(0,1)$ and $(2,1)$ we see that $0<b<1$. Thus the probability of point $P$ not satisfying the condition in the first square is $\frac{1}{2}$, meaning the probability it does is $\frac{1}{2}$.\nHence we want, $\frac{1}{2}\cdot{\frac{1}{2}}+\frac{1}{2}\implies{\boxed{\frac{3}{4}}}$

metric geometry - area

Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.\n<image1>

Answer: $\boxed{\frac{3}{4}}$


Question:
The picture shows 5 cubes from the front. What do they look like from above?
<image1>
<image2>

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

descriptive geometry

Question:
The picture shows 5 cubes from the front. What do they look like from above?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The diagram on the right shows a triangle $A B C$ where two lines are drawn to the opposite sides from each of the two vertices $A$ and $B$. This divides the triangle into nine non-overlapping sections. If eight lines are drawn to the opposite sides, four from $A$ and four from $B$, what is the number of nonoverlapping sections the triangle is divided into?
<image1>

Answer: 25

counting

Question:
The diagram on the right shows a triangle $A B C$ where two lines are drawn to the opposite sides from each of the two vertices $A$ and $B$. This divides the triangle into nine non-overlapping sections. If eight lines are drawn to the opposite sides, four from $A$ and four from $B$, what is the number of nonoverlapping sections the triangle is divided into?
<image1>

Answer: 25


Question: Andrew wants to write the numbers $1,2,3,4,5,6$ and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. Which number should he write in the top circle? <image1>

Answer: 4

Solution: Let the numbers in the circles be $p, q, r, s, t, u$ and $v$, as shown. Since these numbers are 1,2,3,4, 5, 6 and 7 in some order,  $$ p+q+r+s+t+u+v=1+2+3+4+5+6+7=28 $$  ![](https://cdn.mathpix.com/cropped/2023_12_27_23f752c0584f83753c92g-11.jpg?height=229&width=317&top_left_y=708&top_left_x=1555)  Let the sum of the numbers in each line of three circles be $X$. Consider the five lines of three circles. We have  $$ (p+q+t)+(p+r+u)+(p+s+v)+(q+r+s)+(t+u+v)=5 X $$  . Therefore  $$ 3 p+2 q+2 r+2 s+2 t+2 u+2 v=5 X $$  and hence  $$ p+2(p+q+r+s+t+u+v)=5 X $$  . Therefore  $$ 5 X=p+2 \times 28=p+56 $$  , and hence $p+56$ is a multiple of 5 . Since $p$ is one of the numbers $1,2,3,4,5,6$ and 7 , the only number for which this is possible is $p=4$.  Note: This shows that if a solution exists, then $p=4$. It can then be shown that such a solution is possible, for example with $p=4, q=3, r=7, s=2, t=5, u=1, v=6$.

combinatorics

Question: Andrew wants to write the numbers $1,2,3,4,5,6$ and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. Which number should he write in the top circle? <image1>

Answer: 4


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

Answer: 72

metric geometry - area

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

Answer: 72


Question:
In the isosceles triangle $A B C$, points $K$ and $L$ are marked on the equal sides $A B$ and $B C$ respectively so that $A K=K L=L B$ and $K B=A C$.
<image1>
What is the size of angle $A B C$ ?

Choices:
- $36^{\circ}$
- $38^{\circ}$
- $40^{\circ}$
- $42^{\circ}$
- $44^{\circ}$

Solution: Since triangle $A B C$ is isosceles with $A B=B C$ and we are given that $L B=A K$, the other parts of the equal sides must themselves be equal. Hence $L C=B K=A C$. Draw in line $K C$ as shown to form triangles $A C K$ and $L C K$. Since $A K=K L, A C=L C$ and $K C$ is common to both, triangles $A C K$ and $L C K$ are congruent and hence $\angle K A C=\angle C L K$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-246.jpg?height=271&width=252&top_left_y=1806&top_left_x=1573)  Let the size of $\angle L B K$ be $x^{\circ}$. Since $K L=L B$, triangle $K L B$ is isosceles and hence $\angle B K L=x^{\circ}$. Since an exterior angle of a triangle is equal to the sum of the interior opposite angles, $\angle K L C=2 x^{\circ}$ and hence $\angle K A C=2 x^{\circ}$. Since the base angles of an isosceles triangle are equal, $\angle A C L=2 x^{\circ}$. Therefore, since angles in a triangle add to $180^{\circ}$, when we consider triangle $A B C$, we have $x^{\circ}+2 x^{\circ}+2 x^{\circ}=180^{\circ}$ and hence $x=36$. Therefore the size of $\angle A B C$ is $36^{\circ}$.

metric geometry - angle

Question:
In the isosceles triangle $A B C$, points $K$ and $L$ are marked on the equal sides $A B$ and $B C$ respectively so that $A K=K L=L B$ and $K B=A C$.
<image1>
What is the size of angle $A B C$ ?
(A) $36^{\circ}$
(B) $38^{\circ}$
(C) $40^{\circ}$
(D) $42^{\circ}$
(E) $44^{\circ}$


Question: In a rectangle $A B C D$, the points $P, Q, R$ and $S$ are the midpoints of sides $A B, B C, C D$ and $A D$ respectively, and $T$ is the midpoint of the line $R S$. What fraction of the area of $A B C D$ is the triangle $P Q T$ ? <image1>

Choices:
- $\frac{5}{16}$
- $\frac{1}{4}$
- $\frac{1}{5}$
- $\frac{1}{6}$
- $\frac{3}{8}$

Solution: A moment's thought shows that the rhombus $P Q R S$ occupies half the area of the rectangle, and that the triangle $P Q T$ occupies half the area of the rhombus.  Note that the exact position of $T$ on $R S$ is irrelevant.  [Alternatively:  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-010.jpg?height=252&width=300&top_left_y=1887&top_left_x=1432)  Slide $T$ along to $S$ : the area will not be altered because the base and height of the triangle are unchanged. Next slide $P$ along to $A$ and again the area will not be changed. The area of the new triangle is a quarter of the rectangle $A B C D$.]

metric geometry - area

Question:
In a rectangle $A B C D$, the points $P, Q, R$ and $S$ are the midpoints of sides $A B, B C, C D$ and $A D$ respectively, and $T$ is the midpoint of the line $R S$. What fraction of the area of $A B C D$ is the triangle $P Q T$ ? <image1>
(A) $\frac{5}{16}$
(B) $\frac{1}{4}$
(C) $\frac{1}{5}$
(D) $\frac{1}{6}$
(E) $\frac{3}{8}$


Question:
Two square sheets are made up of seethrough and black little squares. Both are placed on top of each other onto the sheet in the middle. Which shape can then still be seen?
<image1>
<image2>

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

descriptive geometry

Question:
Two square sheets are made up of seethrough and black little squares. Both are placed on top of each other onto the sheet in the middle. Which shape can then still be seen?
<image1>
<image2>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
In the diagram below, angle $ABC$ is a right angle.  Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$.  Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$.  Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$.

<image1>

Answer: 148

metric geometry - area

Question:
In the diagram below, angle $ABC$ is a right angle.  Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$.  Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$.  Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$.

<image1>

Answer: 148


Question:
Paula's weather app shows a diagram of the predicted weather and maximum temperatures for the next seven days, as shown. Which of the following represents the corresponding graph of maximum temperatures?
<image1>
<image2>

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

statistics

Question:
Paula's weather app shows a diagram of the predicted weather and maximum temperatures for the next seven days, as shown. Which of the following represents the corresponding graph of maximum temperatures?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Jack makes a cube from 27 small cubes. The small cubes are either grey or white as shown in the diagram. Two small cubes with the same colour are not allowed to be placed next to each other. How many small, white cubes has Jack used?
<image1>

Answer: 13

descriptive geometry

Question:
Jack makes a cube from 27 small cubes. The small cubes are either grey or white as shown in the diagram. Two small cubes with the same colour are not allowed to be placed next to each other. How many small, white cubes has Jack used?
<image1>

Answer: 13


Question:
The lengths of the sides of triangle $X Y Z$ are $X Z=\sqrt{55}$, $X Y=8, Y Z=9$. Find the length of the diagonal $X A$ of the rectangular parallelepiped in the figure.
<image1>

Answer: 10

solid geometry

Question:
The lengths of the sides of triangle $X Y Z$ are $X Z=\sqrt{55}$, $X Y=8, Y Z=9$. Find the length of the diagonal $X A$ of the rectangular parallelepiped in the figure.
<image1>

Answer: 10


Question:
$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR + PS$ is equal to:
<image1>

Choices:
- PQ
- AE
- PT + AT
- AF
- EF

metric geometry - length

Question:
$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR + PS$ is equal to:
<image1>
(A) PQ
(B) AE
(C) PT + AT
(D) AF
(E) EF


Question:
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle.  If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between

<image1>

Choices:
- $4\text{ and }5$
- $5\text{ and }6$
- $6\text{ and }7$
- $7\text{ and }8$
- $8\text{ and }9$

metric geometry - area

Question:
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle.  If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between

<image1>
(A) $4\text{ and }5$
(B) $5\text{ and }6$
(C) $6\text{ and }7$
(D) $7\text{ and }8$
(E) $8\text{ and }9$


Question:
How many quadratic functions $y=a x^{2}+b x+c$ (with $a \neq 0$ ) have graphs that go through at least 3 of the marked points?
<image1>

Answer: 22

analytic geometry

Question:
How many quadratic functions $y=a x^{2}+b x+c$ (with $a \neq 0$ ) have graphs that go through at least 3 of the marked points?
<image1>

Answer: 22


Question:
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)

<image1>

Answer: 28

combinatorics

Question:
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)

<image1>

Answer: 28


Question:
A cube is cut in three directions as shown, to produce eight cuboids (each cut is parallel to one of the faces of the cube). What is the ratio of the total surface area of the eight cuboids to the surface area of the original cube?
<image1>

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

solid geometry

Question:
A cube is cut in three directions as shown, to produce eight cuboids (each cut is parallel to one of the faces of the cube). What is the ratio of the total surface area of the eight cuboids to the surface area of the original cube?
<image1>
(A) 1:1
(B) 4:3
(C) 3:2
(D) 2:1
(E) 4:1


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

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

logic

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


Question:
Cynthia paints each region of the figure in a single color: red, blue or yellow. She paints with different colors the regions that touch each other. In how many different ways can Cyntia paint the figure?
<image1>

Answer: 6

graph theory

Question:
Cynthia paints each region of the figure in a single color: red, blue or yellow. She paints with different colors the regions that touch each other. In how many different ways can Cyntia paint the figure?
<image1>

Answer: 6


Question:
Four circles with radius 1 intersect each other as seen in the diagram. What is the perimeter of the grey area?
<image1>

Choices:
- $\pi$
- $\frac{3 \pi}{2}$
- a number between $\frac{3 \pi}{2}$ and $2 \pi$
- $2 \pi$
- $\pi^{2}$

metric geometry - length

Question:
Four circles with radius 1 intersect each other as seen in the diagram. What is the perimeter of the grey area?
<image1>
(A) $\pi$
(B) $\frac{3 \pi}{2}$
(C) a number between $\frac{3 \pi}{2}$ and $2 \pi$
(D) $2 \pi$
(E) $\pi^{2}$


Question:
Jane, Kate and Lynn go for a walk. Jane walks at the very front, Kate in the middle and Lynn at the very back. Jane weighs $500 \mathrm{~kg}$ more than Kate and Kate weighs $1000 \mathrm{~kg}$ less than Lynn. Which of the following pictures shows Jane, Kate and Lynn in the right order?
<image1>

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

combinatorics

Question:
Jane, Kate and Lynn go for a walk. Jane walks at the very front, Kate in the middle and Lynn at the very back. Jane weighs $500 \mathrm{~kg}$ more than Kate and Kate weighs $1000 \mathrm{~kg}$ less than Lynn. Which of the following pictures shows Jane, Kate and Lynn in the right order?
<image1>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

<image1>

Choices:
- $\frac{1}{6}$
- $\frac{1}{5}$
- $\frac{2}{9}$
- $\frac{1}{3}$
- $\frac{\sqrt{2}}{4}$

metric geometry - area

Question:
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

<image1>
(A) $\frac{1}{6}$
(B) $\frac{1}{5}$
(C) $\frac{2}{9}$
(D) $\frac{1}{3}$
(E) $\frac{\sqrt{2}}{4}$


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

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

descriptive geometry

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


Question:
Each object shown is made up of 7 cubes. Which of $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ can be obtained by rotating $\mathrm{T}$ ?
<image1>

Choices:
- P and R
- Q and S
- only R
- none of them
- P, Q and R

Solution: Stand each object on a table with the central cube uppermost. Then the plan view, from directly above, of $P$, $\mathrm{R}$ and $\mathrm{T}$ is shown in the left diagram, and that of $\mathrm{Q}$ and $\mathrm{S}$ is on the right. In each case the central cube is shown shaded. Mathematicians and scientists say that they have different chirality, that is, they can be considered as left-right-handed versions of the same object.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-066.jpg?height=270&width=440&top_left_y=662&top_left_x=1395)

descriptive geometry

Question:
Each object shown is made up of 7 cubes. Which of $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ can be obtained by rotating $\mathrm{T}$ ?
<image1>
(A) P and R
(B) Q and S
(C) only R
(D) none of them
(E) P, Q and R


Question: Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice's view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.)\n<image1>

Answer: 156

Solution: The region is composed of a $120^{\circ}$ sector of the annulus plus two $60^{\circ}$ sectors with radius 10 , minus two 30-60-90 triangles of side lengths $5,5 \sqrt{3}$, and 10 (see diagram). The area of the annulus sector is $\frac{120}{360} \pi\left(10^2-5^2\right)=25 \pi$, the total area of the two triangles is $2 \cdot \frac{25 \sqrt{3}}{2}=25 \sqrt{3}$, and the total area of the $60^{\circ}$ sectors is $2 \cdot \frac{60}{360} \cdot \pi \cdot 10^2=\frac{100 \pi}{3}$. Adding and subtracting in the right order gives an area of\n$$\n25 \pi-25 \sqrt{3}+\frac{100 \pi}{3}=\frac{175 \pi}{3}-25 \sqrt{3}\n$$\nand thus our final answer is 156 .

metric geometry - area

Question: Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice's view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.)\n<image1>

Answer: 156


Question:
Five sparrows on a rope look in one or the other direction (see diagram). Every sparrow whistles as many times as the number of sparrows he can see in front of him. Azra therefore whistles four times. Then one sparrow turns in the opposite direction and again all sparrows whistle according to the same rule. The second time the sparrows whistle more often in total than the first time. Which sparrow has turned around?
<image1>

Choices:
- Azra
- Bernhard
- Christa
- David
- Elsa

logic

Question:
Five sparrows on a rope look in one or the other direction (see diagram). Every sparrow whistles as many times as the number of sparrows he can see in front of him. Azra therefore whistles four times. Then one sparrow turns in the opposite direction and again all sparrows whistle according to the same rule. The second time the sparrows whistle more often in total than the first time. Which sparrow has turned around?
<image1>
(A) Azra
(B) Bernhard
(C) Christa
(D) David
(E) Elsa


Question:
How many triangles are in this figure? (Some triangles may overlap other triangles.)
<image1>

Answer: 5

combinatorial geometry

Question:
How many triangles are in this figure? (Some triangles may overlap other triangles.)
<image1>

Answer: 5


Question:
Caroline wants to write the numbers $1,2,3,4$ in the square $4 \times 4$ in such a way that every row and every column has each of the numbers. You see how she started. How many of the 4 numbers can be written in place of $x$?
<image1>

Answer: 2

logic

Question:
Caroline wants to write the numbers $1,2,3,4$ in the square $4 \times 4$ in such a way that every row and every column has each of the numbers. You see how she started. How many of the 4 numbers can be written in place of $x$?
<image1>

Answer: 2


Question:
In the addition sum to the right, three digits have been replaced with stars. How big is the sum of the three missing digits?
<image1>

Answer: 0

algebra

Question:
In the addition sum to the right, three digits have been replaced with stars. How big is the sum of the three missing digits?
<image1>

Answer: 0


Question:
In the quadrilateral $A B C D$ the diagonal $B D$ is the bisector of $\angle A B C$ and $A C=B C$. Given $\angle B D C=80^{\circ}$ and $\angle A C B=20^{\circ}, \angle B A D$ is equal to:
<image1>

Choices:
- $90^{\circ}$
- $100^{\circ}$
- $110^{\circ}$
- $120^{\circ}$
- $135^{\circ}$

metric geometry - angle

Question:
In the quadrilateral $A B C D$ the diagonal $B D$ is the bisector of $\angle A B C$ and $A C=B C$. Given $\angle B D C=80^{\circ}$ and $\angle A C B=20^{\circ}, \angle B A D$ is equal to:
<image1>
(A) $90^{\circ}$
(B) $100^{\circ}$
(C) $110^{\circ}$
(D) $120^{\circ}$
(E) $135^{\circ}$


Question: The points $G$ and $I$ are on the circle with centre $H$, and $F I$ is tangent to the circle at $I$. The distances $F G$ and $H I$ are integers, and $F I=F G+6$. The point $G$ lies on the straight line through $F$ and $H$. How many possible values are there for $H I$ ? <image1>

Answer: 6

Solution: Let $x$ be the length $F G$ and let $r$ be the radius. Then $F I=x+6$ and $G H=H I=r$. Angle $F I H$ is a right angle (the tangent and radius are perpendicular) so $F I^{2}+H I^{2}+F H^{2}$, which gives $(x+6)^{2}+r^{2}=(x+r)^{2}$. Expanding this gives $x^{2}+12 x+36+r^{2}=x^{2}+2 r x+r^{2}$, which simplifies to $12 x+36=2 r x$. Halving this gives $6 x+18=r x$, which rearranges to $r=6+\frac{18}{x}$. Since $r$ is an integer, $x$ must be a (positive) factor of 18, namely $1,2,3,6$, 9,18 ; each of these six factors give a different value of $r$ (or $H I$ ) as required.

metric geometry - length

Question: The points $G$ and $I$ are on the circle with centre $H$, and $F I$ is tangent to the circle at $I$. The distances $F G$ and $H I$ are integers, and $F I=F G+6$. The point $G$ lies on the straight line through $F$ and $H$. How many possible values are there for $H I$ ? <image1>

Answer: 6


Question:
Two rectangles are inscribed into a triangle as shown in the diagram. The dimensions of the rectangles are $1 \times 5$ and $2 \times 3$ respectively. How big is the height of the triangle in $A$?
<image1>

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

metric geometry - length

Question:
Two rectangles are inscribed into a triangle as shown in the diagram. The dimensions of the rectangles are $1 \times 5$ and $2 \times 3$ respectively. How big is the height of the triangle in $A$?
<image1>
(A) 3
(B) $\frac{7}{2}$
(C) $\frac{8}{3}$
(D) $\frac{6}{5}$
(E) another number


Question:
The different digits are build using sticks as shown. The ñweightò of a number describes the number of sticks used to build
it. How heavy is the heaviest two digit number?
<image1>

Answer: 14

arithmetic

Question:
The different digits are build using sticks as shown. The ñweightò of a number describes the number of sticks used to build
it. How heavy is the heaviest two digit number?
<image1>

Answer: 14


Question:
Three semi-circles, the diameters of two of which are equal to 4 and of the third to 8, are arranged as seen in the picture. What is the distance from the center $S$ of the bigger semi-circle to the tangent point $T$ of the smaller semi-circles?
<image1>

Choices:
- $6.$
- $\sqrt{32}$
- 5.7
- $\sqrt{40}$
- 5

metric geometry - length

Question:
Three semi-circles, the diameters of two of which are equal to 4 and of the third to 8, are arranged as seen in the picture. What is the distance from the center $S$ of the bigger semi-circle to the tangent point $T$ of the smaller semi-circles?
<image1>
(A) $6.$
(B) $\sqrt{32}$
(C) 5.7
(D) $\sqrt{40}$
(E) 5


Question:
The numbers $1,4,7,10$ and 13 should be written into the squares so that the sum of the three numbers in the horizontal row is equal to the sum of the three numbers in the vertical column. What is the largest possible value of these sums?
<image1>

Answer: 24

algebra

Question:
The numbers $1,4,7,10$ and 13 should be written into the squares so that the sum of the three numbers in the horizontal row is equal to the sum of the three numbers in the vertical column. What is the largest possible value of these sums?
<image1>

Answer: 24


Question:
In the diagram opposite there is an object with 6 triangular faces. On each corner there is a number (two are shown). The sum of the numbers on the corners of each face is the same. What is the sum of all 5 numbers?
<image1>

Answer: 17

algebra

Question:
In the diagram opposite there is an object with 6 triangular faces. On each corner there is a number (two are shown). The sum of the numbers on the corners of each face is the same. What is the sum of all 5 numbers?
<image1>

Answer: 17


Question:
How many distinct triangles can be drawn using three of the dots below as vertices?

<image1>

Answer: 18

combinatorial geometry

Question:
How many distinct triangles can be drawn using three of the dots below as vertices?

<image1>

Answer: 18


Question:
In which picture are there more black Kangaroos than white ones?
<image1>

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

counting

Question:
In which picture are there more black Kangaroos than white ones?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Betty walked around the park once, starting from the marked point in the direction of the arrow. She took 4 pictures. In which order did she take the pictures?
<image1>

Answer: 2143

descriptive geometry

Question:
Betty walked around the park once, starting from the marked point in the direction of the arrow. She took 4 pictures. In which order did she take the pictures?
<image1>

Answer: 2143


Question:
Let $ABCD$ be an isoceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD$. Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$

<image1>

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

metric geometry - length

Question:
Let $ABCD$ be an isoceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD$. Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$

<image1>
(A) $3$
(B) $2+\sqrt{2}$
(C) $1+\sqrt{6}$
(D) $2\sqrt{3}$
(E) $3\sqrt{2}$


Question:
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

<image1>

Answer: 12

combinatorics

Question:
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

<image1>

Answer: 12


Question: One corner of a square is folded to its centre to form an irregular pentagon as shown in the diagram. The area of the square is 1 unit greater than the area of the pentagon. What is the area of the square? <image1>

Answer: 8

Solution: The area of the darker triangle in the diagram is $\frac{1}{8}$ of the area of the whole square and this also represents the difference between the area of the square and the area of the pentagon. Hence $\frac{1}{8}$ of the area of the square is equal to 1 unit and so the area of the whole square is 8 units.

transformation geometry

Question: One corner of a square is folded to its centre to form an irregular pentagon as shown in the diagram. The area of the square is 1 unit greater than the area of the pentagon. What is the area of the square? <image1>

Answer: 8


Question:
The points $N, M$ and $L$ lie on the sides of an equilateral triangle $A B C$ so that $\mathrm{NM} \perp \mathrm{BC}, \mathrm{ML} \perp \mathrm{AB}$ and $\mathrm{LN} \perp \mathrm{AC}$ holds true. The area of the triangle $\mathrm{ABC}$ is $36 \mathrm{~cm}^{2}$. What is the area of the triangle LMN?
<image1>

Choices:
- $9 \mathrm{~cm}^{2}$
- $12 \mathrm{~cm}^{2}$
- $15 \mathrm{~cm}^{2}$
- $16 \mathrm{~cm}^{2}$
- $18 \mathrm{~cm}^{2}$

metric geometry - area

Question:
The points $N, M$ and $L$ lie on the sides of an equilateral triangle $A B C$ so that $\mathrm{NM} \perp \mathrm{BC}, \mathrm{ML} \perp \mathrm{AB}$ and $\mathrm{LN} \perp \mathrm{AC}$ holds true. The area of the triangle $\mathrm{ABC}$ is $36 \mathrm{~cm}^{2}$. What is the area of the triangle LMN?
<image1>
(A) $9 \mathrm{~cm}^{2}$
(B) $12 \mathrm{~cm}^{2}$
(C) $15 \mathrm{~cm}^{2}$
(D) $16 \mathrm{~cm}^{2}$
(E) $18 \mathrm{~cm}^{2}$


Question:
The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively.
<image1>
Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than:

Choices:
- 2a+b
- 2a+c
- 2b+c
- a+2b
- a+b+c

metric geometry - length

Question:
The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively.
<image1>
Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than:
(A) 2a+b
(B) 2a+c
(C) 2b+c
(D) a+2b
(E) a+b+c


Question:
Jenny writes numbers into a $3 \times 3$ table so that the sums of the four numbers in each $2 \times 2$ area of the table are the same. The numbers in three of the cells in the corner can already be seen in the diagram. Which number does she write into the cell in the fourth corner?
<image1>

Answer: 1

algebra

Question:
Jenny writes numbers into a $3 \times 3$ table so that the sums of the four numbers in each $2 \times 2$ area of the table are the same. The numbers in three of the cells in the corner can already be seen in the diagram. Which number does she write into the cell in the fourth corner?
<image1>

Answer: 1


Question:
There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square?
<image1>

Choices:
- $\frac{1}{5}$
- $\frac{1}{4}$
- $\frac{2}{5}$
- $\frac{9}{20}$
- $\frac{1}{2}$

combinatorics

Question:
There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square?
<image1>
(A) $\frac{1}{5}$
(B) $\frac{1}{4}$
(C) $\frac{2}{5}$
(D) $\frac{9}{20}$
(E) $\frac{1}{2}$


Question:
Triangle $\triangle ABC$ in the figure has area $10$.  Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$.  If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is

<image1>

Choices:
- $4$
- $5$
- $6$
- $\frac{5}{3}\sqrt{10}$
- $\text{not uniquely determined}$

metric geometry - area

Question:
Triangle $\triangle ABC$ in the figure has area $10$.  Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$.  If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is

<image1>
(A) $4$
(B) $5$
(C) $6$
(D) $\frac{5}{3}\sqrt{10}$
(E) $\text{not uniquely determined}$


Question:
Valentin draws a zig-zag line inside a rectangle as shown in the diagram. For that he uses the angles $10^{\circ}, 14^{\circ}, 33^{\circ}$ and $26^{\circ}$. How big is angle $\varphi$?
<image1>

Choices:
- $11^{\circ}$
- $12^{\circ}$
- $16^{\circ}$
- $17^{\circ}$
- $33^{\circ}$

metric geometry - angle

Question:
Valentin draws a zig-zag line inside a rectangle as shown in the diagram. For that he uses the angles $10^{\circ}, 14^{\circ}, 33^{\circ}$ and $26^{\circ}$. How big is angle $\varphi$?
<image1>
(A) $11^{\circ}$
(B) $12^{\circ}$
(C) $16^{\circ}$
(D) $17^{\circ}$
(E) $33^{\circ}$


Question:
The solid shown in the diagram has 12 regular pentagonal faces, the other faces being either equilateral triangles or squares. Each pentagonal face is surrounded by 5 square faces and each triangular face is surrounded by 3 square faces. John writes 1 on each triangular face, 5 on each pentagonal face and -1 on each square. What is the total of the numbers written on the solid?
<image1>

Answer: 50

solid geometry

Question:
The solid shown in the diagram has 12 regular pentagonal faces, the other faces being either equilateral triangles or squares. Each pentagonal face is surrounded by 5 square faces and each triangular face is surrounded by 3 square faces. John writes 1 on each triangular face, 5 on each pentagonal face and -1 on each square. What is the total of the numbers written on the solid?
<image1>

Answer: 50


Question:
The diagram shows the ground plan of a room. The adjacent walls are perpendicular to each other. What is the area of the room?
<image1>

Choices:
- $2 a b+a(b-a)$
- $3 a(a+b)-a^{2}$
- $3 a^{2} b$
- $3 a(b-a)+a^{2}$
- $3 a b$

metric geometry - area

Question:
The diagram shows the ground plan of a room. The adjacent walls are perpendicular to each other. What is the area of the room?
<image1>
(A) $2 a b+a(b-a)$
(B) $3 a(a+b)-a^{2}$
(C) $3 a^{2} b$
(D) $3 a(b-a)+a^{2}$
(E) $3 a b$


Question:
The square $A B C D$ has area 80. The points $E, F, G$ and $H$ are on the sides of the square and $\mathrm{AE}=\mathrm{BF}=\mathrm{CG}=\mathrm{DH}$. How big is the area of the grey part, if $\mathrm{AE}=3 \times \mathrm{EB}$?
<image1>

Answer: 25

metric geometry - area

Question:
The square $A B C D$ has area 80. The points $E, F, G$ and $H$ are on the sides of the square and $\mathrm{AE}=\mathrm{BF}=\mathrm{CG}=\mathrm{DH}$. How big is the area of the grey part, if $\mathrm{AE}=3 \times \mathrm{EB}$?
<image1>

Answer: 25


Question:
Ellen wants to decorate the butterfly Which butterfly can she make?
<image1>
using these 6 stickers
<image2>
<image3>

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

combinatorial geometry

Question:
Ellen wants to decorate the butterfly Which butterfly can she make?
<image1>
using these 6 stickers
<image2>
<image3>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
There were five candidates in the school election. After $90 \%$ of the votes had been counted, the preliminary results were as shown on the right. How many students still had a chance of winning the election?
<image1>

Answer: 3

Solution: The 45 votes already cast are $90 \%$ of those available. So the remaining $10 \%$ is $45 \div 9=5$ votes. If Henry wins at least two of the five votes then he is certain to win the election. If India or Jenny win all five of these votes, they would be ahead of Henry. But if Ken or Lena secure five more votes, they would still be behind Henry. Hence only Henry, India and Jenny still have a chance of winning.

algebra

Question:
There were five candidates in the school election. After $90 \%$ of the votes had been counted, the preliminary results were as shown on the right. How many students still had a chance of winning the election?
<image1>

Answer: 3


Question: One of the line segments shown on the grid is the image produced by a rotation of the other line segment. Which of the points $T, U, V$, $W$ could be the centre of such a rotation? <image1>

Choices:
- only $T$
- only $U$
- either of $U$ and $W$
- any of $U, V$ and $W$
- any of $T, U, V$ and $W$

Solution: Label the horizontal line segment $P Q$, and the vertical line segment $R S$. A rotation of $90^{\circ}$ anticlockwise about $U$, or $90^{\circ}$ clockwise about $W$ would map $P Q$ onto $R S$.  When a rotation is performed, the distance of any point from the centre of rotation is preserved. Hence $T$ cannot be a centre because it is a distance of 1 unit from R, but more than 1 unit from $P$ and $Q$. Similarly $V$ is less than 2 units from $R$ and $S$, but more than 2 units  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-137.jpg?height=377&width=451&top_left_y=1830&top_left_x=1348) from $Q$. So only $U$ and $W$ can be centres.

transformation geometry

Question:
One of the line segments shown on the grid is the image produced by a rotation of the other line segment. Which of the points $T, U, V$, $W$ could be the centre of such a rotation? <image1>
(A) only $T$
(B) only $U$
(C) either of $U$ and $W$
(D) any of $U, V$ and $W$
(E) any of $T, U, V$ and $W$


Question:
Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?

<image1>

Choices:
- $\frac{12+3\sqrt{3}}4$
- $\frac{9}{2}$
- $3+\sqrt{3}$
- $\frac{6+3\sqrt{3}}2$
- $6$

metric geometry - area

Question:
Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?

<image1>
(A) $\frac{12+3\sqrt{3}}4$
(B) $\frac{9}{2}$
(C) $3+\sqrt{3}$
(D) $\frac{6+3\sqrt{3}}2$
(E) $6$


Question:
Ten kangaroos stood in a line as shown in the diagram.
<image1>
At a particular moment, two kangaroos standing nose-to-nose exchanged places by jumping past each other. Each of the two kangaroos involved in an exchange continued to face the same way as it did before the exchange. This was repeated until no further exchanges were possible. How many exchanges were made?

Answer: 18

Solution: Label the kangaroos facing right as $\mathrm{K} 1, \mathrm{~K} 2, \mathrm{~K} 3, \mathrm{~K} 4, \mathrm{~K} 5$ and $\mathrm{K} 6$ as shown in the diagram.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-230.jpg?height=163&width=1491&top_left_y=2140&top_left_x=431)  No further exchanges will be possible only when the kangaroos facing right have moved past all the kangaroos facing left. Kangaroos K1, K2 and K3 each have four left-facing kangaroos to move past while kangaroos $\mathrm{K} 4, \mathrm{~K} 5$ and $\mathrm{K} 6$ each have two left-facing kangaroos to move past. Hence there will be $(3 \times 4+3 \times 2)=18$ exchanges made before no further exchanges are possible.

counting

Question:
Ten kangaroos stood in a line as shown in the diagram.
<image1>
At a particular moment, two kangaroos standing nose-to-nose exchanged places by jumping past each other. Each of the two kangaroos involved in an exchange continued to face the same way as it did before the exchange. This was repeated until no further exchanges were possible. How many exchanges were made?

Answer: 18


Question:
We have three horizontal lines and three parallel, sloped lines. Both of the circles shown touch four of the lines. X, Y and Z are the areas of the grey regions. $\mathrm{D}$ is the area of the parallelogram PQRS. At least how many of the areas $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ and $\mathrm{D}$ does one have to know in order to be able to determine the area of the parallelogram $\mathrm{T}$ ?
<image1>

Answer: 1

metric geometry - area

Question:
We have three horizontal lines and three parallel, sloped lines. Both of the circles shown touch four of the lines. X, Y and Z are the areas of the grey regions. $\mathrm{D}$ is the area of the parallelogram PQRS. At least how many of the areas $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ and $\mathrm{D}$ does one have to know in order to be able to determine the area of the parallelogram $\mathrm{T}$ ?
<image1>

Answer: 1


Question:
$\mathrm{M}$ and $\mathrm{N}$ are the midpoints of the equal sides of an isosceles triangle. How big is the area of the quadrilateral (marked?)?
<image1>

Answer: 6

metric geometry - area

Question:
$\mathrm{M}$ and $\mathrm{N}$ are the midpoints of the equal sides of an isosceles triangle. How big is the area of the quadrilateral (marked?)?
<image1>

Answer: 6


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

Answer: 21

graph theory

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

Answer: 21


Question:
Which of the figures below will cover the most dots when laid on the square shown on the right.
<image1>
<image2>

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

combinatorial geometry

Question:
Which of the figures below will cover the most dots when laid on the square shown on the right.
<image1>
<image2>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
With which square do you have to swap the question mark, so that the white area and the black area are the same size?
<image1>
<image2>

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

combinatorial geometry

Question:
With which square do you have to swap the question mark, so that the white area and the black area are the same size?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: Let $\vartriangle ABC$ be equilateral. Two points $D$ and $E$ are on side $BC$ (with order $B, D, E, C$), and satisfy $\angle DAE = 30^o$ . If $BD = 2$ and $CE = 3$, what is $BC$?\n<image1>

Answer: $5+\sqrt{19}$

Solution: Rotate the figure around $A$ by $60^{\circ}$ so that $C$ comes at the place of $B$. Let $B^{\prime}, C^{\prime}, D^{\prime}, E^{\prime}$ be corresponding points of the moved figure. Since $\angle E^{\prime} A D=\angle E^{\prime} A C^{\prime}+\angle C^{\prime} A D=$ $\angle E A C+\angle B A D=30^{\circ}=\angle E A D, E^{\prime} A=E A$ and $D A=D^{\prime} A$, one has $E^{\prime} D=E D$. So $B C=B D+$ $D E+E A$ can be found out if we know $E^{\prime} D$. But $E^{\prime} D=\sqrt{E^{\prime} B^2+B D^2-2 \cdot E^{\prime} B \cdot B D \cdot \cos 120^{\circ}}=$ $\sqrt{19}$, so $B C=2+\sqrt{19}+3=5+\sqrt{19}$.

metric geometry - length

Question: Let $\vartriangle ABC$ be equilateral. Two points $D$ and $E$ are on side $BC$ (with order $B, D, E, C$), and satisfy $\angle DAE = 30^o$ . If $BD = 2$ and $CE = 3$, what is $BC$?\n<image1>

Answer: $5+\sqrt{19}$


Question: Mary had to run to catch the train, got off two stops later and then walked to school. Which of the following speed-time graphs would best represent her journey? <image1>

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

Solution: The order of speeds needs to be train $>$ running $>$ walk. This rules out $A$ and $B$. The train section of the trip needs to have just a single intermediate stop in the middle of the trip. However $C$ shows no intermediate stop, just a slowing, and $D$ shows two intermediate stops. This leaves $E$ as the only feasible solution.

analytic geometry

Question:
Mary had to run to catch the train, got off two stops later and then walked to school. Which of the following speed-time graphs would best represent her journey? <image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Peter rides his bike along a cycle path in a park. He starts at point $S$ and rides in the direction of the arrow. At the first crossing he turns right, then at the next left, and then again to the right and then again to left. Which crossing does he not reach?
<image1>

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

graph theory

Question:
Peter rides his bike along a cycle path in a park. He starts at point $S$ and rides in the direction of the arrow. At the first crossing he turns right, then at the next left, and then again to the right and then again to left. Which crossing does he not reach?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Max wants to complete the jigsaw shown. He has different pieces.
<image1>
Which pieces does he have to use?
<image2>

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

combinatorial geometry

Question:
Max wants to complete the jigsaw shown. He has different pieces.
<image1>
Which pieces does he have to use?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


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

Answer: 6

topology

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

Answer: 6


Question:
The picture shows 2 mushrooms. What is the difference between their heights?
<image1>

Answer: 5

arithmetic

Question:
The picture shows 2 mushrooms. What is the difference between their heights?
<image1>

Answer: 5


Question:
The two externally tangent circles each have a radius of 1 unit. Each circle is tangent to three sides of the rectangle. What is the area of the shaded region? Express your answer in terms of $\pi$.

<image1>

Answer: 8-2\pi

Solution: Each diameter of a circle is 2 units. The rectangle is 2 diameters by 1 diameter, or 4 units by 2 units. Its area is thus 8 square units. Each circle has an area of $1^2\pi=\pi$ square units, so the two circles have a combined area of $2\pi$ square units. The total shaded area is that of the rectangle minus that of the excluded circles, or $\boxed{8-2\pi}$ square units.

metric geometry - area

Question:
The two externally tangent circles each have a radius of 1 unit. Each circle is tangent to three sides of the rectangle. What is the area of the shaded region? Express your answer in terms of $\pi$.

<image1>

Answer: 8-2\pi


Question:
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown.  The shaded area is

<image1>

Answer: 20

metric geometry - area

Question:
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown.  The shaded area is

<image1>

Answer: 20


Question:
Marco's father took a picture of his son in front of the car shown beside. Which of the drawings below could represent this picture?
<image1>
<image2>

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

descriptive geometry

Question:
Marco's father took a picture of his son in front of the car shown beside. Which of the drawings below could represent this picture?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In an equilateral triangle with area 1, we draw the six perpendicular lines from the midpoints of each side to the other two sides as seen in the diagram. How big is the area of the grey hexagon that has been created this way?
<image1>

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

metric geometry - area

Question:
In an equilateral triangle with area 1, we draw the six perpendicular lines from the midpoints of each side to the other two sides as seen in the diagram. How big is the area of the grey hexagon that has been created this way?
<image1>
(A) $\frac{1}{3}$
(B) $\frac{2}{5}$
(C) $\frac{4}{9}$
(D) $\frac{1}{2}$
(E) $\frac{2}{3}$


Question: In the diagram below, the three circles and the three line segments are tangent as shown. Given that the radius of all of the three circles is $1$, compute the area of the triangle.\n<image1>

Answer: $6+4\sqrt{3}$

metric geometry - area

Question: In the diagram below, the three circles and the three line segments are tangent as shown. Given that the radius of all of the three circles is $1$, compute the area of the triangle.\n<image1>

Answer: $6+4\sqrt{3}$


Question:
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

<image1>

Choices:
- $4$
- $12 - 4\sqrt{3}$
- $3\sqrt{3}$
- $4\sqrt{3}$
- $16 - \sqrt{3}$

metric geometry - area

Question:
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

<image1>
(A) $4$
(B) $12 - 4\sqrt{3}$
(C) $3\sqrt{3}$
(D) $4\sqrt{3}$
(E) $16 - \sqrt{3}$


Question:
Two thousand points are given on a circle.  Label one of the points 1.  From this point, count 2 points in the clockwise direction and label this point 2.  From the point labeled 2, count 3 points in the clockwise direction and label this point 3.  (See figure.)  Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used.  Some of the points on the circle will have more than one label and some points will not have a label.  What is the smallest integer that labels the same point as 1993?

<image1>

Answer: 118

algebra

Question:
Two thousand points are given on a circle.  Label one of the points 1.  From this point, count 2 points in the clockwise direction and label this point 2.  From the point labeled 2, count 3 points in the clockwise direction and label this point 3.  (See figure.)  Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used.  Some of the points on the circle will have more than one label and some points will not have a label.  What is the smallest integer that labels the same point as 1993?

<image1>

Answer: 118


Question:
Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle?
<image1>

Answer: 3

descriptive geometry

Question:
Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle?
<image1>

Answer: 3


Question: Eve has taken 2 bananas to school. At first she changed each of them into 4 apples, later on she exchanged each apple into 3 mandarins. How many mandarins has Eve got? <image1>

Choices:
- $2+4+3$
- $2 \cdot 4+3$
- $2+4 \cdot 3$
- $2 \cdot 4 \cdot 3$
- $2+4-3$

arithmetic

Question:
Eve has taken 2 bananas to school. At first she changed each of them into 4 apples, later on she exchanged each apple into 3 mandarins. How many mandarins has Eve got? <image1>
(A) $2+4+3$
(B) $2 \cdot 4+3$
(C) $2+4 \cdot 3$
(D) $2 \cdot 4 \cdot 3$
(E) $2+4-3$


Question: The Pentagon $P Q R S T$ is divided into four triangles with equal perimeters. The triangle $P Q R$ is equilateral. $P T U, S U T$ and $R S U$ are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon $P Q R S T$ to the perimeter of the triangle $P Q R$? <image1>

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

Solution: Let $2 a$ be the side of equilateral triangle $P Q R$. Then $P Q R$ has a perimeter of $6 a$ and $P U$ has a length $a$.  Therefore, in order for the perimeter of triangle $P T U$ to be $6 a, P T$ must be $\frac{5}{2} a$.  Since the isosceles triangles are congruent $S T=P U$ and $R S=P T$. So the perimeter of the pentagon $P Q R S T$ is $2 a+2 a+\frac{5}{2} a+a+\frac{5}{2} a=10 a$. Therefore the ratio wanted is $10 a: 6 a$, that is $5: 3$.

metric geometry - length

Question:
The Pentagon $P Q R S T$ is divided into four triangles with equal perimeters. The triangle $P Q R$ is equilateral. $P T U, S U T$ and $R S U$ are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon $P Q R S T$ to the perimeter of the triangle $P Q R$? <image1>
(A) $2: 1$
(B) $3: 2$
(C) $4: 3$
(D) $5: 3$
(E) $5: 2$


Question:
In the diagram QSR is a straight line. $\angle \mathrm{QPS}=12^{\circ}$ and $\mathrm{PQ}=\mathrm{PS}=\mathrm{RS}$. How big is $\angle \mathrm{QPR}$?
<image1>

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

metric geometry - angle

Question:
In the diagram QSR is a straight line. $\angle \mathrm{QPS}=12^{\circ}$ and $\mathrm{PQ}=\mathrm{PS}=\mathrm{RS}$. How big is $\angle \mathrm{QPR}$?
<image1>
(A) $36^{\circ}$
(B) $42^{\circ}$
(C) $54^{\circ}$
(D) $60^{\circ}$
(E) $84^{\circ}$


Question: An ant walks along the sides of an equilateral triangle (see diagram). Its velocity is $5 \mathrm{~cm} / \mathrm{min}$ along the first side, $15 \mathrm{~cm} / \mathrm{min}$ along the second and $20 \mathrm{~cm} / \mathrm{min}$ along the third. With which average velocity in $\mathrm{cm} / \mathrm{min}$ does the ant walk once around the entire triangle? <image1>

Choices:
- $10$
- $\frac{80}{11}$
- $\frac{180}{19}$
- $15$
- $\frac{40}{3}$

algebra

Question:
An ant walks along the sides of an equilateral triangle (see diagram). Its velocity is $5 \mathrm{~cm} / \mathrm{min}$ along the first side, $15 \mathrm{~cm} / \mathrm{min}$ along the second and $20 \mathrm{~cm} / \mathrm{min}$ along the third. With which average velocity in $\mathrm{cm} / \mathrm{min}$ does the ant walk once around the entire triangle? <image1>
(A) $10$
(B) $\frac{80}{11}$
(C) $\frac{180}{19}$
(D) $15$
(E) $\frac{40}{3}$


Question:
In the figure $A B C D$ is a rectangle and $P Q R S$ a square. The area of the grey part is half as big as the area of ABCD. How long is the side PX?
<image1>

Answer: 1

metric geometry - length

Question:
In the figure $A B C D$ is a rectangle and $P Q R S$ a square. The area of the grey part is half as big as the area of ABCD. How long is the side PX?
<image1>

Answer: 1


Question:
A $4 \times 4$ square is made up of the two pieces shown. Which of the following $4 \times 4$ squares cannot be made this way?
<image1>
<image2>

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

combinatorial geometry

Question:
A $4 \times 4$ square is made up of the two pieces shown. Which of the following $4 \times 4$ squares cannot be made this way?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the figure there are nine regions inside the five circles. All of the numbers from 1 to 9 are written in the regions, one to each region, so that the sum of the numbers inside each circle is 11 .
Which number must be written in the region with the question mark?
<image1>

Answer: 5

Solution: The sum of the digits from 1 to 9 is 45 . There are 5 circles and the sum of the numbers in each circle is 11 , which gives a total of $5 \times 11=55$. However, this counts the four numbers which are in more than one circle twice and the sum of these numbers is $55-45=10$. Therefore the  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-114.jpg?height=257&width=577&top_left_y=1890&top_left_x=1276) shared numbers are 1, 2,3,4 since this is the only combination of four different digits that add to 10 . Now in order to give 11 as the total in each circle, 1 must go in a circle with two other digits and let $P$ be the other number in a circle with two other digits. Suppose that $P$ is replaced by the number 2 . Then the region with the question mark must contain the number 8 . But 3 and 8 must then be in one of the outer circles and this is not possible if we are to use each digit only once. Similarly, if $P$ is replaced by the number 3 , then the region with the question mark must contain the number 7. But 4 and 7 must then be in one of the outer circles, and again, this isn't possible. Therefore $P$ must be replaced by the number 4 . Hence 6 is in the region with the question mark and the diagram shows that there is an arrangement as desired.

combinatorics

Question:
In the figure there are nine regions inside the five circles. All of the numbers from 1 to 9 are written in the regions, one to each region, so that the sum of the numbers inside each circle is 11 .
Which number must be written in the region with the question mark?
<image1>

Answer: 5


Question:
Stan has five toys: a ball, a set of blocks, a game, a puzzle and a car. He puts each toy on a different shelf of the bookcase. The ball is higher than the blocks and lower than the car. The game is directly above the ball. On which shelf can the puzzle not be placed?
<image1>

Answer: 3

logic

Question:
Stan has five toys: a ball, a set of blocks, a game, a puzzle and a car. He puts each toy on a different shelf of the bookcase. The ball is higher than the blocks and lower than the car. The game is directly above the ball. On which shelf can the puzzle not be placed?
<image1>

Answer: 3


Question: The diagram shows a spiral of isosceles triangles. The largest angle in each of the triangles is $100^{\circ}$. The grey triangle is number 0 . Each of the following triangles (numbered 1, 2, 3, ...) join by one edge to the previous one, as shown. As you can see triangle 3 only partially covers triangle 0. What is the number of the first triangle that exactly covers triangle 0 ? <image1>

Answer: 18

Solution: If we consider the anticlockwise angle between the edge of triangle 0 and the corresponding edge of each succeeding triangle this will increase by $100^{\circ}$ with each triangle, so for triangle $n$ this angle is $100 n^{\circ}$. For triangle $n$ exactly to cover triangle $0,100 n$ must be a multiple of 360 , so $5 n$ must be a multiple of 18 . This first occurs when $n=18$.

transformation geometry

Question: The diagram shows a spiral of isosceles triangles. The largest angle in each of the triangles is $100^{\circ}$. The grey triangle is number 0 . Each of the following triangles (numbered 1, 2, 3, ...) join by one edge to the previous one, as shown. As you can see triangle 3 only partially covers triangle 0. What is the number of the first triangle that exactly covers triangle 0 ? <image1>

Answer: 18


Question:
Four cogs are connected to each other as shown in the picture. The first has 30 teeth, the second 15, the third 60 and the fourth 10 . How many turns will the last cog make for each full turn of the first cog?
<image1>

Answer: 3

transformation geometry

Question:
Four cogs are connected to each other as shown in the picture. The first has 30 teeth, the second 15, the third 60 and the fourth 10 . How many turns will the last cog make for each full turn of the first cog?
<image1>

Answer: 3


Question:
The road from Anna's to Mary's house is $16 \mathrm{~km}$ long. The road from Mary's to John's house is $20 \mathrm{~km}$ long. The road from the crossing to Mary's house is $9 \mathrm{~km}$ long. How long is the road from Anna's to John's house?
<image1>

Choices:
- $7 \mathrm{~km}$
- $9 \mathrm{~km}$
- $11 \mathrm{~km}$
- $16 \mathrm{~km}$
- $18 \mathrm{~km}$

arithmetic

Question:
The road from Anna's to Mary's house is $16 \mathrm{~km}$ long. The road from Mary's to John's house is $20 \mathrm{~km}$ long. The road from the crossing to Mary's house is $9 \mathrm{~km}$ long. How long is the road from Anna's to John's house?
<image1>
(A) $7 \mathrm{~km}$
(B) $9 \mathrm{~km}$
(C) $11 \mathrm{~km}$
(D) $16 \mathrm{~km}$
(E) $18 \mathrm{~km}$


Question:
The faces of a cube are painted black, white or grey. Each face is only painted one colour and opposite faces are painted the same colour. Which of the following is a possible net for the cube?
<image1>

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

Solution: Each of the nets shown has two faces of each colour. The question tells us that the two faces are opposite each other, so they cannot have an edge in common. This eliminates all the nets except net $\mathrm{B}$.

solid geometry

Question:
The faces of a cube are painted black, white or grey. Each face is only painted one colour and opposite faces are painted the same colour. Which of the following is a possible net for the cube?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The shortest way from Atown to Cetown is through Betown. Going back by this road from Cetown to Atown, we first find the signposts on the left side of the road. Further on we find the road signs on the right side of the road. How far is it from Betown to Atown?
<image1>

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

metric geometry - length

Question:
The shortest way from Atown to Cetown is through Betown. Going back by this road from Cetown to Atown, we first find the signposts on the left side of the road. Further on we find the road signs on the right side of the road. How far is it from Betown to Atown?
<image1>
(A) $1 \mathrm{~km}$
(B) $2 \mathrm{~km}$
(C) $3 \mathrm{~km}$
(D) $4 \mathrm{~km}$
(E) $5 \mathrm{~km}$


Question: Consider the two touching semicircles with radius 1 and their diameters $A B$ and $C D$ respectively that are parallel to each other. The extensions of the two diameters are also tangents to the respective other semicircle (see diagram). How big is the square of the length $A D$ ? <image1>

Choices:
- 16
- $8+4 \sqrt{3}$
- 12
- 9
- $5+2 \sqrt{3}$

metric geometry - length

Question:
Consider the two touching semicircles with radius 1 and their diameters $A B$ and $C D$ respectively that are parallel to each other. The extensions of the two diameters are also tangents to the respective other semicircle (see diagram). How big is the square of the length $A D$ ? <image1>
(A) 16
(B) $8+4 \sqrt{3}$
(C) 12
(D) 9
(E) $5+2 \sqrt{3}$


Question:
The clock shown has a rectangular clock face, the hands however move as usual in a constant circular pattern. How big is the distance $x$ of the digits 1 and 2 (in $\mathrm{cm}$ ), if the distance between the numbers 8 and 10 is given as $12 \mathrm{~cm}$?
<image1>

Choices:
- $3 \sqrt{3}$
- $2 \sqrt{3}$
- $4 \sqrt{3}$
- $2+\sqrt{3}$
- $12-3 \sqrt{3}$

metric geometry - length

Question:
The clock shown has a rectangular clock face, the hands however move as usual in a constant circular pattern. How big is the distance $x$ of the digits 1 and 2 (in $\mathrm{cm}$ ), if the distance between the numbers 8 and 10 is given as $12 \mathrm{~cm}$?
<image1>
(A) $3 \sqrt{3}$
(B) $2 \sqrt{3}$
(C) $4 \sqrt{3}$
(D) $2+\sqrt{3}$
(E) $12-3 \sqrt{3}$


Question:
In this square there are 9 dots. The distance between the points is always the same. You can draw a square by joining 4 points. How many different sizes can such squares have?
<image1>

Answer: 3

counting

Question:
In this square there are 9 dots. The distance between the points is always the same. You can draw a square by joining 4 points. How many different sizes can such squares have?
<image1>

Answer: 3


Question:
The billiard ball meets the board under $45^{\circ}$ as shown. Which pocket will it fall into?
<image1>

Choices:
- $A$
- $B$
- $C$
- $D$
- Neither of the pockets

transformation geometry

Question:
The billiard ball meets the board under $45^{\circ}$ as shown. Which pocket will it fall into?
<image1>
(A) $A$
(B) $B$
(C) $C$
(D) $D$
(E) Neither of the pockets


Question:
A marble of radius 15 is rolled into a cone-shaped hole. It fits in perfectly. From the side the cone looks like an equilateral triangle. How deep is the hole?
<image1>

Answer: 45

solid geometry

Question:
A marble of radius 15 is rolled into a cone-shaped hole. It fits in perfectly. From the side the cone looks like an equilateral triangle. How deep is the hole?
<image1>

Answer: 45


Question:
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:
<image1>

Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?

Answer: 5

algebra

Question:
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:
<image1>

Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?

Answer: 5


Question:
A $3 \times 3 \times 3$ cube weighs 810 grams. If we drill three holes through it as shown, each of which is a $1 \times 1 \times 3$ rectangular parallelepiped, the weight of the remaining solid is:
<image1>

Choices:
- $540 \mathrm{~g}$
- $570 \mathrm{~g}$
- $600 \mathrm{~g}$
- $630 \mathrm{~g}$
- $660 \mathrm{~g}$

solid geometry

Question:
A $3 \times 3 \times 3$ cube weighs 810 grams. If we drill three holes through it as shown, each of which is a $1 \times 1 \times 3$ rectangular parallelepiped, the weight of the remaining solid is:
<image1>
(A) $540 \mathrm{~g}$
(B) $570 \mathrm{~g}$
(C) $600 \mathrm{~g}$
(D) $630 \mathrm{~g}$
(E) $660 \mathrm{~g}$


Question:
Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for his sister. How many pieces were there in the entire bar of chocolate?
<image1>

Answer: 40

metric geometry - area

Question:
Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for his sister. How many pieces were there in the entire bar of chocolate?
<image1>

Answer: 40


Question:
How many triangles can be drawn with vertices in the 18 points shown in the figure?
<image1>

Answer: 711

combinatorics

Question:
How many triangles can be drawn with vertices in the 18 points shown in the figure?
<image1>

Answer: 711


Question:
We used metal rods to build this nice ensemble. We know there are 61 octagons. How many rods are there?
<image1>

Answer: 446

algebra

Question:
We used metal rods to build this nice ensemble. We know there are 61 octagons. How many rods are there?
<image1>

Answer: 446


Question:
In the diagram shown, you should follow the arrows to get from A to B. How many different ways are there that fulfill this condition?
<image1>

Answer: 16

combinatorics

Question:
In the diagram shown, you should follow the arrows to get from A to B. How many different ways are there that fulfill this condition?
<image1>

Answer: 16


Question:
In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is
<image1>

Choices:
- $2$
- $3 - \frac{\sqrt{3}}{2}$
- $\frac{11}{5}$
- $\sqrt{5}$
- $\frac{9}{4}$

metric geometry - area

Question:
In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is
<image1>
(A) $2$
(B) $3 - \frac{\sqrt{3}}{2}$
(C) $\frac{11}{5}$
(D) $\sqrt{5}$
(E) $\frac{9}{4}$


Question: The flag of Kangaria is a rectangle with side-lengths in the ratio $3: 5$. The flag is divided into four rectangles of equal area as shown. What is the ratio of the length of the shorter sides of the white rectangle to the length of its longer sides? <image1>

Choices:
- $1: 3$
- $1: 4$
- $2: 7$
- $3: 10$
- $4: 15$

Solution: Let the length of the white rectangle be $x$ and its height $y$. Then the height of the flag is $3 y$ and hence its width is $5 y$. The four rectangles which make up the flag are equal in area, so we have $3 y \times 5 y=4 x y$. This simplifies to $15 y=4 x$ (since $y$ is non-zero) and hence $y: x=4: 15$.

metric geometry - length

Question:
The flag of Kangaria is a rectangle with side-lengths in the ratio $3: 5$. The flag is divided into four rectangles of equal area as shown. What is the ratio of the length of the shorter sides of the white rectangle to the length of its longer sides? <image1>
(A) $1: 3$
(B) $1: 4$
(C) $2: 7$
(D) $3: 10$
(E) $4: 15$


Question:
Maria wants there to be a knife to the right of every plate and a fork to the left of it. In order to get the right order she always swaps one fork with one knife. What is the minimum number of swaps necessary?
<image1>

Answer: 2

graph theory

Question:
Maria wants there to be a knife to the right of every plate and a fork to the left of it. In order to get the right order she always swaps one fork with one knife. What is the minimum number of swaps necessary?
<image1>

Answer: 2


Question:
A cuboid is formed from 3 pieces (see picture). Each piece is made from 4 cubes of the same colour. What shape does the white piece have?
<image1>
<image2>

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

combinatorial geometry

Question:
A cuboid is formed from 3 pieces (see picture). Each piece is made from 4 cubes of the same colour. What shape does the white piece have?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The diagram shows the plan of a room. Adjoining walls are perpendicular to each other and the lengths of some of the walls are shown. What is the length of the perimeter of the room? <image1>

Choices:
- $3 a+4 b$
- $3 a+8 b$
- $6 a+4 b$
- $6 a+6 b$
- $6 a+8 b$

Solution: One long wall has length $b+2 b+b=4 b$ and the perpendicular long wall has length $a+a+a=3 a$. So the length of the perimeter is $6 a+8 b$.

metric geometry - length

Question:
The diagram shows the plan of a room. Adjoining walls are perpendicular to each other and the lengths of some of the walls are shown. What is the length of the perimeter of the room? <image1>
(A) $3 a+4 b$
(B) $3 a+8 b$
(C) $6 a+4 b$
(D) $6 a+6 b$
(E) $6 a+8 b$


Question:
Picture $X$ is paired with picture $Y$. Which of the following pictures is paired with picture $G$ ?
<image1>
<image2>

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

combinatorial geometry

Question:
Picture $X$ is paired with picture $Y$. Which of the following pictures is paired with picture $G$ ?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E