Question:
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.]; What is the ratio of the area of triangle $A$ to the area of triangle $B$?

<image1>

Choices:
- 1
- m/n
- n/m
- 2m/n
- 2n/m

metric geometry - area

Question:
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.]; What is the ratio of the area of triangle $A$ to the area of triangle $B$?

<image1>
(A) 1
(B) m/n
(C) n/m
(D) 2m/n
(E) 2n/m


Question:
Chantal has placed numbers in two of the nine cells (see diagram). She wants to place the numbers 1, 2, 3 in every row and every column exactly once. How big is the sum of the two numbers in the grey cells?
<image1>

Answer: 4

logic

Question:
Chantal has placed numbers in two of the nine cells (see diagram). She wants to place the numbers 1, 2, 3 in every row and every column exactly once. How big is the sum of the two numbers in the grey cells?
<image1>

Answer: 4


Question: In the grid on the right, there are eight kangaroos. A kangaroo may jump into any empty square. Find the least number of the kangaroos which have to jump into an empty square so that in each row and column there are exactly two kangaroos. <image1>

Answer: 1

Solution: Once one kangaroo hops as shown, there are exactly 2 kangaroos in every row and column.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-044.jpg?height=291&width=300&top_left_y=668&top_left_x=1346)

logic

Question: In the grid on the right, there are eight kangaroos. A kangaroo may jump into any empty square. Find the least number of the kangaroos which have to jump into an empty square so that in each row and column there are exactly two kangaroos. <image1>

Answer: 1


Question:
The pattern shown in the diagram is constructed using semicircles. Each semicircle has a diameter that lies on the horizontal axis shown and has one of the black dots at either end. The distance between each pair of adjacent black dots is $1 \mathrm{~cm}$. The area, in $\mathrm{cm}^{2}$, of the pattern that is shaded in grey is $\frac{1}{8} k \pi$. What is the value of $k$ ?
<image1>

Answer: 121

Solution: Take each shaded semi-annulus that is below the line and reflect it in the line then move it one centimetre to the left. A shaded semicircle of diameter $11 \mathrm{~cm}$ is obtained. Therefore the whole shaded area is $\frac{121}{8} \pi$. Hence the value of $k$ is 121 .

metric geometry - area

Question:
The pattern shown in the diagram is constructed using semicircles. Each semicircle has a diameter that lies on the horizontal axis shown and has one of the black dots at either end. The distance between each pair of adjacent black dots is $1 \mathrm{~cm}$. The area, in $\mathrm{cm}^{2}$, of the pattern that is shaded in grey is $\frac{1}{8} k \pi$. What is the value of $k$ ?
<image1>

Answer: 121


Question:
In triangle $ABC$, $\measuredangle CBA=72^\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is

<image1>

Choices:
- $\frac{1}{5}$
- $\frac{1}{4}$
- $\frac{1}{3}$
- $\frac{2}{5}$
- $\text{none of these}$

metric geometry - area

Question:
In triangle $ABC$, $\measuredangle CBA=72^\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is

<image1>
(A) $\frac{1}{5}$
(B) $\frac{1}{4}$
(C) $\frac{1}{3}$
(D) $\frac{2}{5}$
(E) $\text{none of these}$


Question:
n number of buttons are placed evenly around a circle. The buttons are labelled clockwise in order with the numbers 1 to $n$. The button with the number 7 is exactly opposite the button with the number 23. How big is $n$?
<image1>

Answer: 32

algebra

Question:
n number of buttons are placed evenly around a circle. The buttons are labelled clockwise in order with the numbers 1 to $n$. The button with the number 7 is exactly opposite the button with the number 23. How big is $n$?
<image1>

Answer: 32


Question:
In which picture are there half as many circles as triangles and twice as many squares as triangles?
<image1>

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

counting

Question:
In which picture are there half as many circles as triangles and twice as many squares as triangles?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The rings shown are partially interlinked. How long is the longest chain built this way which also contains the thick light ring?
<image1>

Answer: 5

topology

Question:
The rings shown are partially interlinked. How long is the longest chain built this way which also contains the thick light ring?
<image1>

Answer: 5


Question: Werner wants to write a number at each vertex and on each edge of the rhombus shown. He wants the sum of the numbers at the two vertices at the ends of each edge to be equal to the number written on that edge. What number should he write on the edge marked with the question mark? <image1>

Answer: 12

Solution: The number on each edge is equal to the sum of the two numbers at the vertices at the ends of the edge. Therefore, the sum of the numbers on two opposite edges will equal the sum of the numbers at all four vertices. Therefore $8+13=9+?$ and hence $?=12$.

algebra

Question: Werner wants to write a number at each vertex and on each edge of the rhombus shown. He wants the sum of the numbers at the two vertices at the ends of each edge to be equal to the number written on that edge. What number should he write on the edge marked with the question mark? <image1>

Answer: 12


Question:
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.

<image1>

Answer: 8

combinatorial geometry

Question:
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.

<image1>

Answer: 8


Question:
The square $A B C D$ has side length $3 \mathrm{~cm}$. The points $M$ and $N$, which lie on the sides $A D$ and $A B$ respectively, are joined to the corner $C$. That way the square is split up into three parts with equal area. How long is the line segment DM?
<image1>

Choices:
- $0.5 \mathrm{~cm}$
- $1 \mathrm{~cm}$
- $1.5 \mathrm{~cm}$
- $2 \mathrm{~cm}$
- $2.5 \mathrm{~cm}$

metric geometry - length

Question:
The square $A B C D$ has side length $3 \mathrm{~cm}$. The points $M$ and $N$, which lie on the sides $A D$ and $A B$ respectively, are joined to the corner $C$. That way the square is split up into three parts with equal area. How long is the line segment DM?
<image1>
(A) $0.5 \mathrm{~cm}$
(B) $1 \mathrm{~cm}$
(C) $1.5 \mathrm{~cm}$
(D) $2 \mathrm{~cm}$
(E) $2.5 \mathrm{~cm}$


Question:
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following?
<image1>

Choices:
- 20
- 20.5
- 21
- 21.5
- 22

metric geometry - length

Question:
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following?
<image1>
(A) 20
(B) 20.5
(C) 21
(D) 21.5
(E) 22


Question: What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated.\n<image1>

Answer: 8

Solution: It is easy to show that 8 is possible. First each tile has area 4 and total area is 36 so we can have at most 9 tiles. We will show that 9 is not possible. Color the 6 × 6 grid in a checkboard pattern. Then there are an 18 white squares. Each tile will cover either 1 or 3 white squares. Hence 9 tiles will cover an odd number of white squares. This isn't possible so the maximum is 8.

combinatorial geometry

Question: What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated.\n<image1>

Answer: 8


Question:
The diagram shows the floor plan of Renate's house. Renate enters her house from the terrace (Terrasse) and walks through every door of the house exactly once. Which room does she end up in?
<image1>

Answer: 2

graph theory

Question:
The diagram shows the floor plan of Renate's house. Renate enters her house from the terrace (Terrasse) and walks through every door of the house exactly once. Which room does she end up in?
<image1>

Answer: 2


Question:
A piece of string is lying on the table. It is partially covered by three coins as seen in the figure. Under each coin the string is equally likely to pass over itself like this: <image1>
or like this: <image2>. What is the probability that the string is knotted after its ends are pulled?
<image3>

Choices:
- $\frac{1}{2}$
- $\frac{1}{4}$
- $\frac{1}{8}$
- $\frac{3}{4}$
- $\frac{3}{8}$

topology

Question:
A piece of string is lying on the table. It is partially covered by three coins as seen in the figure. Under each coin the string is equally likely to pass over itself like this: <image1>
or like this: <image2>. What is the probability that the string is knotted after its ends are pulled?
<image3>
(A) $\frac{1}{2}$
(B) $\frac{1}{4}$
(C) $\frac{1}{8}$
(D) $\frac{3}{4}$
(E) $\frac{3}{8}$


Question:
In the picture on the left we see three dice on top of each other. The sum of the points on opposite sides of the dice is 7 as usual. The sum of the points of areas that face each other is always 5. How many points are on the area marked $\mathrm{X}$?
<image1>

Answer: 6

logic

Question:
In the picture on the left we see three dice on top of each other. The sum of the points on opposite sides of the dice is 7 as usual. The sum of the points of areas that face each other is always 5. How many points are on the area marked $\mathrm{X}$?
<image1>

Answer: 6


Question:
The Kangaroo Hotel has 30 floors numbered from 1 to 30 and each floor has 20 rooms numbered from 1 to 20. The code to enter the room is formed by joining the floor number with the room number, in that order. But this code can be confusing, as shown in the picture. Note that the code 101 is not confusing, as it can only refer to floor 10 and room 1 and never to floor 1 and room 1, as it has the code 11. How many codes are confusing, including the one in the figure?
<image1>

Answer: 18

counting

Question:
The Kangaroo Hotel has 30 floors numbered from 1 to 30 and each floor has 20 rooms numbered from 1 to 20. The code to enter the room is formed by joining the floor number with the room number, in that order. But this code can be confusing, as shown in the picture. Note that the code 101 is not confusing, as it can only refer to floor 10 and room 1 and never to floor 1 and room 1, as it has the code 11. How many codes are confusing, including the one in the figure?
<image1>

Answer: 18


Question:
How big is the angle indicated with a question mark?
<image1>

Choices:
- $10^{\circ}$
- $20^{\circ}$
- $30^{\circ}$
- $40^{\circ}$
- $50^{\circ}$

metric geometry - angle

Question:
How big is the angle indicated with a question mark?
<image1>
(A) $10^{\circ}$
(B) $20^{\circ}$
(C) $30^{\circ}$
(D) $40^{\circ}$
(E) $50^{\circ}$


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

Answer: 75

metric geometry - area

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

Answer: 75


Question:
Five boys share 10 bags of marbles between themselves. Everyone gets exactly two bags:
<image1>
Alex gets 5 marbles, Bob 7, Charles 9 and Dennis 15. Eric gets the two bags that are left over. How many marbles does he get?

Answer: 19

arithmetic

Question:
Five boys share 10 bags of marbles between themselves. Everyone gets exactly two bags:
<image1>
Alex gets 5 marbles, Bob 7, Charles 9 and Dennis 15. Eric gets the two bags that are left over. How many marbles does he get?

Answer: 19


Question:
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

<image1>

Choices:
- $120$
- $180$
- $240$
- $360$
- $480$

solid geometry

Question:
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

<image1>
(A) $120$
(B) $180$
(C) $240$
(D) $360$
(E) $480$


Question:
After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

<image1>

Answer: 24

statistics

Question:
After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

<image1>

Answer: 24


Question:
Petra crafts a piece of jewellery out of two black and two white hearts. The hearts have areas of $1 \mathrm{~cm}^{2}, 4 \mathrm{~cm}^{2}, 9 \mathrm{~cm}^{2}$ and $16 \mathrm{~cm}^{2}$ respectively. She places the hearts on top of each other as shown in the diagram and glues them together. How big is the total area of the visible black parts?
<image1>

Choices:
- $9 \mathrm{~cm}^{2}$
- $10 \mathrm{~cm}^{2}$
- $11 \mathrm{~cm}^{2}$
- $12 \mathrm{~cm}^{2}$
- $13 \mathrm{~cm}^{2}$

metric geometry - area

Question:
Petra crafts a piece of jewellery out of two black and two white hearts. The hearts have areas of $1 \mathrm{~cm}^{2}, 4 \mathrm{~cm}^{2}, 9 \mathrm{~cm}^{2}$ and $16 \mathrm{~cm}^{2}$ respectively. She places the hearts on top of each other as shown in the diagram and glues them together. How big is the total area of the visible black parts?
<image1>
(A) $9 \mathrm{~cm}^{2}$
(B) $10 \mathrm{~cm}^{2}$
(C) $11 \mathrm{~cm}^{2}$
(D) $12 \mathrm{~cm}^{2}$
(E) $13 \mathrm{~cm}^{2}$


Question: Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^o$ and $(n + 1)^o$ with the tangent at $P$, where $n$ is a positive integer less than $90$. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ($P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?\n<image1>

Answer: 28

Solution: Note that we want the path drawn out by the lasers to come back to $P$ in as few steps as possible. Observe that if a laser is fired with an angle of $n$ degrees from the tangent, then the number of points it creates on the circle is $\frac{180}{\operatorname{gcd}(180, n)}$. (Consider the regular polygon created by linking all the points that show up on the circle-if the center of the circle is $\mathrm{O}$, and the vertices are numbered $V_1, V_2, \ldots, V_k$, the angle $\angle V_1 O V_2$ is equal to $2 \operatorname{gcd}(180, n)$, so there are a total of $\frac{360}{2 \operatorname{gcd}(180, n)}$ sides).\nNow, we consider the case with both $n$ and $n+1$. Note that we wish to minimize the value $\frac{180}{\operatorname{gcd}(180, n)}+$ $\frac{180}{\operatorname{gcd}(180, n+1)}$, or maximize both $\operatorname{gcd}(180, n)$ and $\operatorname{gcd}(180, n+1)$. Note that since $n$ and $n+1$ are relatively prime and $180=(4)(9)(5)$, the expression is maximized when $\operatorname{gcd}(180, n)=20$ and $\operatorname{god}(180, n+1)=9$ (or vice versa). This occurs when $n=80$. Plugging this into our expression, we have that the number of points that show up from the laser fired at 80 degrees is $\frac{180}{20}=9$ and the number of points that appear from the laser fired at 81 degrees is $\frac{180}{9}=20$. However, since both have a point that shows up at $P$ (and no other overlapping points since $\operatorname{gcd}(9,20)=1$ ), we see that the answer is $20+9-1=28$.

transformation geometry

Question: Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^o$ and $(n + 1)^o$ with the tangent at $P$, where $n$ is a positive integer less than $90$. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ($P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?\n<image1>

Answer: 28


Question:
There are 5 boxes and each box contains some cards labeled K, M, H, P, T, as shown below. Peter wants to remove cards out of each box so that at the end each box contained only one card, and different boxes contained cards with different letters. Which card remains in the first box?
<image1>

Choices:
- It is impossible to do this
- T
- M
- H
- P

logic

Question:
There are 5 boxes and each box contains some cards labeled K, M, H, P, T, as shown below. Peter wants to remove cards out of each box so that at the end each box contained only one card, and different boxes contained cards with different letters. Which card remains in the first box?
<image1>
(A) It is impossible to do this
(B) T
(C) M
(D) H
(E) P


Question:
The big equilateral triangle consists of 36 small equilateral triangles which each have an area of $1 \mathrm{~cm}^{2}$. Determine the area of $A B C$.
<image1>

Choices:
- $11 \mathrm{~cm}^{2}$
- $12 \mathrm{~cm}^{2}$
- $13 \mathrm{~cm}^{2}$
- $14 \mathrm{~cm}^{2}$
- $15 \mathrm{~cm}^{2}$

combinatorial geometry

Question:
The big equilateral triangle consists of 36 small equilateral triangles which each have an area of $1 \mathrm{~cm}^{2}$. Determine the area of $A B C$.
<image1>
(A) $11 \mathrm{~cm}^{2}$
(B) $12 \mathrm{~cm}^{2}$
(C) $13 \mathrm{~cm}^{2}$
(D) $14 \mathrm{~cm}^{2}$
(E) $15 \mathrm{~cm}^{2}$


Question:
Max colours in the squares of the grid, so that one third of all squares are blue and one half of all squares are yellow. The rest he colours in red. How many squares does he have to colour in red?
<image1>

Answer: 3

arithmetic

Question:
Max colours in the squares of the grid, so that one third of all squares are blue and one half of all squares are yellow. The rest he colours in red. How many squares does he have to colour in red?
<image1>

Answer: 3


Question:
Maddie has a paper ribbon of length $36 \mathrm{~cm}$. She divides it into four rectangles of different lengths. She draws two lines joining the centres of two adjacent rectangles as shown.
<image1>
What is the sum of the lengths of the lines that she draws?

Choices:
- $18 \mathrm{~cm}$
- $17 \mathrm{~cm}$
- $20 \mathrm{~cm}$
- $19 \mathrm{~cm}$
- It depends upon the sizes of the rectangles

Solution: Let the lengths of the four rectangles be $p \mathrm{~cm}, q \mathrm{~cm}, r \mathrm{~cm}$ and $s \mathrm{~cm}$ with $p+q+r+s=36$. The lines Maddie draws join the centres of two pairs of rectangles and hence have total length $\left(\frac{1}{2} p+\frac{1}{2} q\right) \mathrm{cm}+\left(\frac{1}{2} r+\frac{1}{2} s\right) \mathrm{cm}=\frac{1}{2}(p+q+r+s) \mathrm{cm}$. Therefore the sum of the lengths of the lines she draws is $\frac{1}{2} \times 36 \mathrm{~cm}=18 \mathrm{~cm}$.

transformation geometry

Question:
Maddie has a paper ribbon of length $36 \mathrm{~cm}$. She divides it into four rectangles of different lengths. She draws two lines joining the centres of two adjacent rectangles as shown.
<image1>
What is the sum of the lengths of the lines that she draws?
(A) $18 \mathrm{~cm}$
(B) $17 \mathrm{~cm}$
(C) $20 \mathrm{~cm}$
(D) $19 \mathrm{~cm}$
(E) It depends upon the sizes of the rectangles


Question:
Bob folds a piece of paper, then punches a hole into the paper and unfolds it again. The unfolded paper then looks like this:
<image1>
Along which dotted line has Bob folded the paper beforehand?
<image2>

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

transformation geometry

Question:
Bob folds a piece of paper, then punches a hole into the paper and unfolds it again. The unfolded paper then looks like this:
<image1>
Along which dotted line has Bob folded the paper beforehand?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Renate wants to glue together a number of ordinary dice (whose number of points on opposite sides always adds up to 7) to form a "dicebar" as shown. Doing this she only wants to glue sides together with an equal number of points. She wants to make sure that the sum of all points on the non-glued sides equals 2012. How many dice does she have to glue together?
<image1>

Choices:
- 70
- 71
- 142
- 143
- It is impossible to obtain exactly 2012 points on the non-glued together sides.

arithmetic

Question:
Renate wants to glue together a number of ordinary dice (whose number of points on opposite sides always adds up to 7) to form a "dicebar" as shown. Doing this she only wants to glue sides together with an equal number of points. She wants to make sure that the sum of all points on the non-glued sides equals 2012. How many dice does she have to glue together?
<image1>
(A) 70
(B) 71
(C) 142
(D) 143
(E) It is impossible to obtain exactly 2012 points on the non-glued together sides.


Question:
The picture shows 3 gears with a black gear tooth on each. Which picture shows the correct position of the black teeth after the small gear has turned a full turn clockwise?
<image1>
<image2>

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

transformation geometry

Question:
The picture shows 3 gears with a black gear tooth on each. Which picture shows the correct position of the black teeth after the small gear has turned a full turn clockwise?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A farmer's rectangular field is partitioned into  $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
<image1>

Answer: 84

combinatorics

Question:
A farmer's rectangular field is partitioned into  $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
<image1>

Answer: 84


Question:
Andrew, the absent-minded mountain climber, climbed a mountain range, with a profile as shown in Figure 1 from point $P$ to point $Q$. From time to time he had to turn back to find bits of his equipment that he had dropped. The graph of the height $H$ of his position at time $t$ is shown in Figure 2 to the same scale as Figure 1. How many times did he turn back?
<image1>

Answer: 3

Solution: Andrew turned back three times in total as marked on the right-hand diagram.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-031.jpg?height=360&width=1300&top_left_y=2033&top_left_x=499)

statistics

Question:
Andrew, the absent-minded mountain climber, climbed a mountain range, with a profile as shown in Figure 1 from point $P$ to point $Q$. From time to time he had to turn back to find bits of his equipment that he had dropped. The graph of the height $H$ of his position at time $t$ is shown in Figure 2 to the same scale as Figure 1. How many times did he turn back?
<image1>

Answer: 3


Question: In the figure below, quadrilateral $CDEG$ is a square with $CD = 3$, and quadrilateral $BEFH$ is a rectangle. If $BE = 5$, how many units is $BH$? Express your answer as a mixed number. <image1>

Answer: \frac{9}{5}

Solution:
Let $J$ be the intersection of $\overline{BE}$ and $\overline{GC}$. [asy]
unitsize(5mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
pair A=(0,0), B=(3,0), C=(6,0), D=(9,0), Ep=(9,3), G=(6,3), K=(33/5,9/5);
pair F0=bisectorpoint(B,2*Ep-B), H0=bisectorpoint(Ep,2*B-Ep);
pair H=extension(B,H0,A,G);
pair F=extension(Ep,F0,A,G);
pair J=extension(B,Ep,G,C);
draw(H--B--Ep--F--A--D--Ep--G--C);
draw(G--K);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,S);
label("$D$",D,S);
label("$E$",Ep,E);
label("$F$",F,N);
label("$G$",G,NW);
label("$H$",H,NW);
label("$J$",J,NW);
label("$K$",K,SE);[/asy]

Observe that $BD=\sqrt{BE^2-DE^2}=\sqrt{5^2-3^2}=4$ units.  By the similarity of triangles $BCJ$ and $BDE$, we have  \[
\frac{CJ}{BC}=\frac{DE}{BD},
\] which after substituting becomes \[
\frac{CJ}{4-3}=\frac{3}{4}.
\] We solve to find $CJ=\frac{3}{4}$, which implies that $GJ=3-\frac{3}{4}=\frac{9}{4}$.  Applying the Pythagorean theorem to triangle $GJE$, we find $EJ=\sqrt{3^2+\left(\frac{9}{4}\right)^2}=\frac{15}{4}$.  Define $K$ to be the foot of the perpendicular from $G$ to side $EJ$.  By the similarity of triangles $GKJ$ and $EGJ$, we have  \[
\frac{GK}{GJ}=\frac{EG}{EJ} \implies
\frac{GK}{\frac{9}{4}}=\frac{3}{\frac{15}{4}},
\] which we solve to find $GK=\frac{9}{5}$.  Since $GKBH$ is a rectangle, $BH=GK=\boxed{\frac{9}{5}}$ units.

metric geometry - length

Question: In the figure below, quadrilateral $CDEG$ is a square with $CD = 3$, and quadrilateral $BEFH$ is a rectangle. If $BE = 5$, how many units is $BH$? Express your answer as a mixed number. <image1>

Answer: \frac{9}{5}


Question:
A $5 \times 5$ square is covered with $1 \times 1$ tiles. The design on each tile is made up of three dark triangles and one light triangle (see diagram). The triangles of neighbouring tiles always have the same colour where they join along an edge. The border of the large square is made of dark and light triangles. What is the smallest number of dark triangles that could be amongst them?
<image1>

Answer: 5

combinatorial geometry

Question:
A $5 \times 5$ square is covered with $1 \times 1$ tiles. The design on each tile is made up of three dark triangles and one light triangle (see diagram). The triangles of neighbouring tiles always have the same colour where they join along an edge. The border of the large square is made of dark and light triangles. What is the smallest number of dark triangles that could be amongst them?
<image1>

Answer: 5


Question:
Four of the following five pictures show pieces of the graph of the same quadratic function. Which piece does not belong?
<image1>

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

analytic geometry

Question:
Four of the following five pictures show pieces of the graph of the same quadratic function. Which piece does not belong?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
A large cube is made from 64 small cubes. The 5 visible faces of the large cube are green, the bottom face is red. How many of the small cubes have 3 green faces?
<image1>

Answer: 4

counting

Question:
A large cube is made from 64 small cubes. The 5 visible faces of the large cube are green, the bottom face is red. How many of the small cubes have 3 green faces?
<image1>

Answer: 4


Question:
In $1960$ only $5\%$ of the working adults in Carlin City worked at home. By $1970$ the "at-home" work force increased to $8\%$. In $1980$ there were approximately $15\%$ working at home, and in $1990$ there were $30\%$. The graph that best illustrates this is

<image1>

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

statistics

Question:
In $1960$ only $5\%$ of the working adults in Carlin City worked at home. By $1970$ the "at-home" work force increased to $8\%$. In $1980$ there were approximately $15\%$ working at home, and in $1990$ there were $30\%$. The graph that best illustrates this is

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


Question: Janet enters all the digits from 1 to 9 in the cells of a $3 \times 3$ table, so that each cell contains one digit. She has already entered 1,2,3 and 4, as shown. Two numbers are considered to be 'neighbours' if their cells share an edge. After entering all the numbers, she notices that the sum of the neighbours of 9 is 15 . What is the sum of the neighbours of 8 ? <image1>

Answer: 27

Solution: The sum of the neighbours of 9 is 15 . If 9 were to be placed in the central cell, its neighbours would be $5,6,7$ and 8 with sum 26 so 9 must be placed in one of the cells on the perimeter of the table. So the neighbours of 9 will be the numbers in the middle cell and the two corner cells which are in either the same row or the same column of the table. The largest sum of two such corner cells is $3+4=7$ so, for the sum of the neighbours of 9 to be 15, the number in the middle cell cannot be smaller than 8 . However, the middle square cannot be larger than 8 since we already know 9 is in a perimeter cell. Therefore the number in the middle cell is 8 and its neighbours are 5, 6, 7 and 9 with sum 27.

logic

Question: Janet enters all the digits from 1 to 9 in the cells of a $3 \times 3$ table, so that each cell contains one digit. She has already entered 1,2,3 and 4, as shown. Two numbers are considered to be 'neighbours' if their cells share an edge. After entering all the numbers, she notices that the sum of the neighbours of 9 is 15 . What is the sum of the neighbours of 8 ? <image1>

Answer: 27


Question:
How many triangles can be seen in the picture on the right? (Be careful! A triangle can be also be made by joining several smaller triangles together.)
<image1>

Answer: 10

counting

Question:
How many triangles can be seen in the picture on the right? (Be careful! A triangle can be also be made by joining several smaller triangles together.)
<image1>

Answer: 10


Question: In the $6 \times 8$ grid shown, 24 cells are not intersected by either diagonal. When the diagonals of a $6 \times 10$ grid are drawn, how many cells are not intersected by either diagonal? <image1>

Answer: 32

Solution: The $6 \times 10$ grid can be divided into four $3 \times 5$ grids, each intersected by only one diagonal line as shown.  Each time the diagonal crosses a grid line, it enters a new cell.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-163.jpg?height=159&width=263&top_left_y=2399&top_left_x=1590)  From a start point in the top left corner of the grid, the line crosses two horizontal grid lines and four vertical grid lines to reach the bottom right corner. On the $3 \times 5$ grid, the line does not pass through any points at which the grid lines intersect. The number of cells in the $3 \times 5$ grid that the line intersects is $1+2+4=7$. Hence the total number of cells that are not intersected is $6 \times 10-4 \times 7=32$.

analytic geometry

Question: In the $6 \times 8$ grid shown, 24 cells are not intersected by either diagonal. When the diagonals of a $6 \times 10$ grid are drawn, how many cells are not intersected by either diagonal? <image1>

Answer: 32


Question:
A clock has three hands in different lengths (for seconds, minutes and hours). We don't know the length of each hand but we know that the clock shows the correct time. At 12:55:30 the hands are in the positions shown on the right. What does the clockface look like at 8:10:00?
<image1>
<image2>

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

algebra

Question:
A clock has three hands in different lengths (for seconds, minutes and hours). We don't know the length of each hand but we know that the clock shows the correct time. At 12:55:30 the hands are in the positions shown on the right. What does the clockface look like at 8:10:00?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Three circles of radius $ s$ are drawn in the first quadrant of the $ xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $ x$-axis, and the third is tangent to the first circle and the $ y$-axis. A circle of radius $ r > s$ is tangent to both axes and to the second and third circles. What is $ r/s$?
<image1>

Answer: 9

metric geometry - length

Question:
Three circles of radius $ s$ are drawn in the first quadrant of the $ xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $ x$-axis, and the third is tangent to the first circle and the $ y$-axis. A circle of radius $ r > s$ is tangent to both axes and to the second and third circles. What is $ r/s$?
<image1>

Answer: 9


Question:
The sides of the rectangle $A B C D$ are parallel to the co-ordinate axis. The rectangle lies below the $\mathrm{x}$-axis and to the right of the $\mathrm{y}$-axis, as shown in the diagram. For each of the points A, B, C, D the quotient (y-coordinate):(x-coordinate) is calculated. For which point will you obtain the smallest quotient?
<image1>

Choices:
- A
- B
- C
- D
- It depends on the position of the rectangle and its side lengths.

analytic geometry

Question:
The sides of the rectangle $A B C D$ are parallel to the co-ordinate axis. The rectangle lies below the $\mathrm{x}$-axis and to the right of the $\mathrm{y}$-axis, as shown in the diagram. For each of the points A, B, C, D the quotient (y-coordinate):(x-coordinate) is calculated. For which point will you obtain the smallest quotient?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) It depends on the position of the rectangle and its side lengths.


Question:
The ratio of the radii of two concentric circles is $1: 3$. The line $A C$ a diameter of the biggest circle. A chord $B C$ of the big circle touches the small circle (see diagram). The line $A B$ has length 12. How big is the radius of the big circle?
<image1>

Answer: 18

metric geometry - length

Question:
The ratio of the radii of two concentric circles is $1: 3$. The line $A C$ a diameter of the biggest circle. A chord $B C$ of the big circle touches the small circle (see diagram). The line $A B$ has length 12. How big is the radius of the big circle?
<image1>

Answer: 18


Question:
Each side of each triangle in the diagram is painted either blue, green or red. Four of the sides are already painted. Which colour can the line marked "x" have, if each triangle must have all sides in different colours?
<image1>

Choices:
- only green
- only red
- only blue
- either red or blue
- The question cannot be solved.

logic

Question:
Each side of each triangle in the diagram is painted either blue, green or red. Four of the sides are already painted. Which colour can the line marked "x" have, if each triangle must have all sides in different colours?
<image1>
(A) only green
(B) only red
(C) only blue
(D) either red or blue
(E) The question cannot be solved.


Question:
The upper coin rolls without sliding around the fixed lower coin. Which position will the two coins have afterwards?
<image1>
<image2>

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

transformation geometry

Question:
The upper coin rolls without sliding around the fixed lower coin. Which position will the two coins have afterwards?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: A marble with radius $15 \mathrm{~cm}$ fits exactly under a cone as shown in the diagram. The slant height of the cone is equal to the diameter of its base. What is the height of the cone in $\mathrm{cm}$ ? <image1>

Answer: 45

Solution: Because the slant height of the cone is the same as the diameter of its base, the cross-section of the cone is an equilateral triangle, as shown. The cross-section of the sphere is the incircle of the triangle and has radius $15 \mathrm{~cm}$. By the symmetry of the figure $\angle P O R=360^{\circ} \div 6=60^{\circ}$, and hence, triangle $P O R$ has angles $90^{\circ}, 60^{\circ}$ and $30^{\circ}$ (so it is half of an equilateral triangle). Hence $O R$ is twice $O P$ and  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-138.jpg?height=314&width=350&top_left_y=2219&top_left_x=1479) hence is $30 \mathrm{~cm}$. Since $O T=O R$, it follows that $P T$ is $15+30=45 \mathrm{~cm}$. This is the height of the cone.  [Alternatively: You may know that the medians of a triangle intersect at one third of their heights, so $O P=15 \mathrm{~cm}$ is one third of the height.]

solid geometry

Question: A marble with radius $15 \mathrm{~cm}$ fits exactly under a cone as shown in the diagram. The slant height of the cone is equal to the diameter of its base. What is the height of the cone in $\mathrm{cm}$ ? <image1>

Answer: 45


Question: A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make. Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state?\n<image1>

Answer: 7/3

Solution: Note, we can just move the empty space to one of the spots adjacent to it. We win when the empty space is bottom right. By symmetry, this is only a function of distance. Note, $d(1) = 1/2(1) + 1/2d(2), d(2) = d(1) + 1 \implies d(1) = 1/2 + 1/2d(1) + 1/2 \implies d(1) = 2, d(2) = 3.$ There's a $2/3$ chance we're at $d(1)$ and a $1/3$ chance we're at $d(2).$ So, the expected number of moves is $2/3 \cdot 2 + 1/3\cdot 3 = \fbox{7/3}.$

combinatorics

Question: A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make. Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state?\n<image1>

Answer: 7/3


Question:
Two circles are inscribed into an $11 \mathrm{~cm}$ long and $7 \mathrm{~cm}$ wide rectangle so that they each touch three sides of the rectangle. How big is the distance between the centres of the two circles?
<image1>

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

metric geometry - length

Question:
Two circles are inscribed into an $11 \mathrm{~cm}$ long and $7 \mathrm{~cm}$ wide rectangle so that they each touch three sides of the rectangle. How big is the distance between the centres of the two circles?
<image1>
(A) $1 \mathrm{~cm}$
(B) $2 \mathrm{~cm}$
(C) $3 \mathrm{~cm}$
(D) $4 \mathrm{~cm}$
(E) $5 \mathrm{~cm}$


Question:
Zilda will use six equal cubes and two different rectangular blocks to form the structure beside with eight faces. Before gluing the pieces, she will paint each one entirely and calculated that she will need 18 liters of paint (the color does not matter). How many liters of paint would she use if she painted the whole structure only after gluing the parts?
<image1>

Answer: 11.5

solid geometry

Question:
Zilda will use six equal cubes and two different rectangular blocks to form the structure beside with eight faces. Before gluing the pieces, she will paint each one entirely and calculated that she will need 18 liters of paint (the color does not matter). How many liters of paint would she use if she painted the whole structure only after gluing the parts?
<image1>

Answer: 11.5


Question: The diagram shows a triangle and three circles whose centres are at the vertices of the triangle. The area of the triangle is $80 \mathrm{~cm}^{2}$ and each of the circles has radius $2 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded area? <image1>

Choices:
- 76
- $80-2 \pi$
- $40-4 \pi$
- $80-\pi$
- $78 \pi$

Solution: The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area $\frac{1}{2} \times \pi \times 2^{2}=\frac{4 \pi}{2}=2 \pi$. So the grey area is $(80-2 \pi) \mathrm{cm}^{2}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-105.jpg?height=206&width=343&top_left_y=1416&top_left_x=1459)

metric geometry - area

Question:
The diagram shows a triangle and three circles whose centres are at the vertices of the triangle. The area of the triangle is $80 \mathrm{~cm}^{2}$ and each of the circles has radius $2 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded area? <image1>
(A) 76
(B) $80-2 \pi$
(C) $40-4 \pi$
(D) $80-\pi$
(E) $78 \pi$


Question:
On a standard die the sum of the numbers on opposite faces is always 7. Two identical standard dice are shown in the figure. How many dots could there be on the non-visible right-hand face (marked with "?")?
<image1>

Choices:
- only 5
- only 2
- either 2 or 5
- either 1, 2, 3 or 5
- either 2,3 or 5

solid geometry

Question:
On a standard die the sum of the numbers on opposite faces is always 7. Two identical standard dice are shown in the figure. How many dots could there be on the non-visible right-hand face (marked with "?")?
<image1>
(A) only 5
(B) only 2
(C) either 2 or 5
(D) either 1, 2, 3 or 5
(E) either 2,3 or 5


Question: Carl tries to divide the large shape of squares into smaller pieces using only copies of the T-piece and the F-piece shown on the right. (Pieces may be turned over or around.) What is the smallest possible number of the T-pieces that he can achieve? <image1>

Answer: 2

Solution: F-pieces are made up from 4 small squares and Tpieces from 3. So 22 squares could be covered either by $4 \mathrm{Fs}$ and 2 Ts or by $1 \mathrm{~F}$ and 6 Ts. These diagrams show how Carl might have done it both ways, but the smallest number of T-pieces is 2 . ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-011.jpg?height=228&width=462&top_left_y=300&top_left_x=1316)

combinatorial geometry

Question: Carl tries to divide the large shape of squares into smaller pieces using only copies of the T-piece and the F-piece shown on the right. (Pieces may be turned over or around.) What is the smallest possible number of the T-pieces that he can achieve? <image1>

Answer: 2


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

Answer: 2309415687

algebra

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

Answer: 2309415687


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

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

logic

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


Question:
The picture shows a mouse and a piece of cheese. The mouse is only allowed to move to the neighbouring fields in the direction of the arrows. How many paths are there from the mouse to the cheese?
<image1>

Answer: 6

combinatorics

Question:
The picture shows a mouse and a piece of cheese. The mouse is only allowed to move to the neighbouring fields in the direction of the arrows. How many paths are there from the mouse to the cheese?
<image1>

Answer: 6


Question:
Let $ ABCD$ be a rhombus with $ AC=16$ and $ BD=30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$?
<image1>

Answer: 7

metric geometry - length

Question:
Let $ ABCD$ be a rhombus with $ AC=16$ and $ BD=30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$?
<image1>

Answer: 7


Question:
A rectangular section was cut out of a rectangular block as shown in the diagram. Find the decrease percentage of the surface area.
<image1>

Choices:
- Less than $12.5 \%$
- $12.5 \%$
- More than $12.5 \%$, but less than $25 \%$
- $25 \%$
- More than $25 \%$

solid geometry

Question:
A rectangular section was cut out of a rectangular block as shown in the diagram. Find the decrease percentage of the surface area.
<image1>
(A) Less than $12.5 \%$
(B) $12.5 \%$
(C) More than $12.5 \%$, but less than $25 \%$
(D) $25 \%$
(E) More than $25 \%$


Question: A bush has 10 branches. Each branch has either 5 leaves only or 2 leaves and 1 flower. Which of the following could be the total number of leaves the bush has? <image1>

Choices:
- 45
- 39
- 37
- 31
- None of A to D

Solution: The maximum number of leaves the bush could have is $10 \times 5=50$. Each branch that has two leaves and a flower instead of five leaves reduces the number of leaves the bush has by three. Therefore the total number of leaves the bush has is of the form $50-3 n$ where $n$ is the number of branches with two leaves and a flower. It is straightforward to check that none of options A to D has that form and so the answer is $\mathrm{E}$.

algebra

Question:
A bush has 10 branches. Each branch has either 5 leaves only or 2 leaves and 1 flower. Which of the following could be the total number of leaves the bush has? <image1>
(A) 45
(B) 39
(C) 37
(D) 31
(E) None of A to D


Question:
A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

<image1>

Choices:
- 1:1
- 6:5
- 3:2
- 2:1
- 3:1

metric geometry - area

Question:
A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

<image1>
(A) 1:1
(B) 6:5
(C) 3:2
(D) 2:1
(E) 3:1


Question:
Veronika wears five rings as shown. How many, different ways are there for her to take off the rings one by one?
<image1>

Answer: 20

combinatorics

Question:
Veronika wears five rings as shown. How many, different ways are there for her to take off the rings one by one?
<image1>

Answer: 20


Question: A terrace is covered with square tiles of different sizes. The smallest tile has a perimeter of $80 \mathrm{~cm}$. A snake lay down along the edges of the tiles (see diagram). How long is the snake? <image1>

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

combinatorial geometry

Question:
A terrace is covered with square tiles of different sizes. The smallest tile has a perimeter of $80 \mathrm{~cm}$. A snake lay down along the edges of the tiles (see diagram). How long is the snake? <image1>
(A) $380 \mathrm{~cm}$
(B) $400 \mathrm{~cm}$
(C) $420 \mathrm{~cm}$
(D) $440 \mathrm{~cm}$
(E) $1680 \mathrm{~cm}$


Question: In the diagram, the two triangles shown have parallel bases. What is the ratio of the area of the smaller triangle to the area of the larger triangle? <image1>

Answer: \frac{4}{25}

Solution:
Because of our parallel bases, we can see that corresponding angles of the triangles must be congruent. Therefore, by AA similarity, we see that the two triangles are similar.

If two similar triangles have side ratios of $r : 1,$ the ratio of their areas must be $r^2 : 1.$ In our diagram, we see that the ratio of the sides of the smaller triangle to the sides of the larger triangle is $\frac{4\text{ cm}}{10\text{ cm}} = \frac{2}{5}.$ Therefore, the ratio of the areas is the square of that, or $\left(\frac{2}{5}\right)^2 = \boxed{\frac{4}{25}}.$

metric geometry - area

Question: In the diagram, the two triangles shown have parallel bases. What is the ratio of the area of the smaller triangle to the area of the larger triangle? <image1>

Answer: \frac{4}{25}


Question:
Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3,4,12, and 13, respectively, and $\measuredangle CBA$ is a right angle. The area of the quadrilateral is

<image1>

Answer: 36

metric geometry - area

Question:
Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3,4,12, and 13, respectively, and $\measuredangle CBA$ is a right angle. The area of the quadrilateral is

<image1>

Answer: 36


Question:
This network of eight equilateral triangles can be folded to form a regular octahedron. To construct a magic octahedron, replace the letters $A, B, C, D$, and $E$ with the numbers 2, 4, 6,7, and 8 (without repetition) so that each sum of the four numbers on the four faces that share a vertex were the same. On your magic octahedron, what does $B+D$ equal?
<image1>

Answer: 6

solid geometry

Question:
This network of eight equilateral triangles can be folded to form a regular octahedron. To construct a magic octahedron, replace the letters $A, B, C, D$, and $E$ with the numbers 2, 4, 6,7, and 8 (without repetition) so that each sum of the four numbers on the four faces that share a vertex were the same. On your magic octahedron, what does $B+D$ equal?
<image1>

Answer: 6


Question:
The five balls weigh $30 \mathrm{~g}$, $50 \mathrm{~g}, 50 \mathrm{~g}, 50 \mathrm{~g}$ and $80 \mathrm{~g}$. Which of the balls weighs $30 \mathrm{~g}$ ?
<image1>

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

algebra

Question:
The five balls weigh $30 \mathrm{~g}$, $50 \mathrm{~g}, 50 \mathrm{~g}, 50 \mathrm{~g}$ and $80 \mathrm{~g}$. Which of the balls weighs $30 \mathrm{~g}$ ?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the middle of the big diagram one piece is missing and should be replaced. You are only allowed to do this by connecting light-grey lines with light-grey lines, dark-grey lines with dark-grey lines and black lines with black lines. Which piece fits?
<image1>
<image2>

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

combinatorial geometry

Question:
In the middle of the big diagram one piece is missing and should be replaced. You are only allowed to do this by connecting light-grey lines with light-grey lines, dark-grey lines with dark-grey lines and black lines with black lines. Which piece fits?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the diagram on the right the following can be seen: a straight line, which is the common tangent of two touching circles with radius 1, and a square with one edge on the straight line and the other vertices one on each of the two circles. How big is the side length of the square?
<image1>

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

metric geometry - length

Question:
In the diagram on the right the following can be seen: a straight line, which is the common tangent of two touching circles with radius 1, and a square with one edge on the straight line and the other vertices one on each of the two circles. How big is the side length of the square?
<image1>
(A) $\frac{2}{5}$
(B) $\frac{1}{4}$
(C) $\frac{1}{\sqrt{2}}$
(D) $\frac{1}{\sqrt{5}}$
(E) $\frac{1}{2}$


Question:
In this diagram $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which is inscribed equilateral triangle $DEF$. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. Then:
<image1>

Choices:
- $b=\frac{a+c}{2}$
- $b=\frac{a-c}{2}$
- $a=\frac{b-c}{2}$
- $a=\frac{b+c}{2}$
- $\text{none of these}$

metric geometry - angle

Question:
In this diagram $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which is inscribed equilateral triangle $DEF$. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. Then:
<image1>
(A) $b=\frac{a+c}{2}$
(B) $b=\frac{a-c}{2}$
(C) $a=\frac{b-c}{2}$
(D) $a=\frac{b+c}{2}$
(E) $\text{none of these}$


Question:
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

<image1>

Answer: 16

combinatorial geometry

Question:
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

<image1>

Answer: 16


Question:
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card?

<image1>

Choices:
- $14$
- $10\sqrt{2}$
- $16$
- $12\sqrt{2}$
- $18$

metric geometry - area

Question:
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card?

<image1>
(A) $14$
(B) $10\sqrt{2}$
(C) $16$
(D) $12\sqrt{2}$
(E) $18$


Question: A $4\times 6$ grid should be cut along the black lines into several identical shapes. No piece is to be left over. Into which of the following shapes is it not possible to cut this grid in this way? <image1> <image2>

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

combinatorial geometry

Question:
A $4\times 6$ grid should be cut along the black lines into several identical shapes. No piece is to be left over. Into which of the following shapes is it not possible to cut this grid in this way? <image1> <image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: The square $A B C D$ consists of four congruent rectangles arranged around a central square. The perimeter of each of the rectangles is $40 \mathrm{~cm}$. What is the area of the square $A B C D$ ? <image1>

Choices:
- $400 \mathrm{~cm}^{2}$
- $200 \mathrm{~cm}^{2}$
- $160 \mathrm{~cm}^{2}$
- $120 \mathrm{~cm}^{2}$
- $80 \mathrm{~cm}^{2}$

Solution: Let the length of each of the rectangles be $x \mathrm{~cm}$ and the width be $y \mathrm{~cm}$. The perimeter of each of the rectangles is $40 \mathrm{~cm}$ and hence $2 x+2 y=40$. Therefore $x+y=20$. From the diagram we can see that the length of each side of the square $A B C D$ is $(x+y) \mathrm{cm}$. Therefore the square $A B C D$ has side length $20 \mathrm{~cm}$. Hence the area of $A B C D$ is $(20 \times 20) \mathrm{cm}^{2}=400 \mathrm{~cm}^{2}$.

metric geometry - area

Question:
The square $A B C D$ consists of four congruent rectangles arranged around a central square. The perimeter of each of the rectangles is $40 \mathrm{~cm}$. What is the area of the square $A B C D$ ? <image1>
(A) $400 \mathrm{~cm}^{2}$
(B) $200 \mathrm{~cm}^{2}$
(C) $160 \mathrm{~cm}^{2}$
(D) $120 \mathrm{~cm}^{2}$
(E) $80 \mathrm{~cm}^{2}$


Question: The rectangle shown is divided into six squares. The length of the sides of the smallest square is 1 . What is the length of the sides of the largest square? <image1>

Answer: 7

metric geometry - length

Question: The rectangle shown is divided into six squares. The length of the sides of the smallest square is 1 . What is the length of the sides of the largest square? <image1>

Answer: 7


Question:
Baris has a few dominoes as shown in the picture. He wants to lay them in a line according to the rules of dominoes, that is that two dominoes can only be laid together if the neighbouring squares have the same number of dots in them. What is the biggest number of these dominoes that he can lay in a single line?
<image1>

Answer: 5

graph theory

Question:
Baris has a few dominoes as shown in the picture. He wants to lay them in a line according to the rules of dominoes, that is that two dominoes can only be laid together if the neighbouring squares have the same number of dots in them. What is the biggest number of these dominoes that he can lay in a single line?
<image1>

Answer: 5


Question:
Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$.  Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm.

<image1>

Answer: 193

metric geometry - length

Question:
Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$.  Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm.

<image1>

Answer: 193


Question:
Isosceles $ \triangle ABC$ has a right angle at $ C$.  Point $ P$ is inside $ \triangle ABC$, such that $ PA = 11, PB = 7,$ and $ PC = 6$.  Legs $ \overline{AC}$ and $ \overline{BC}$ have length $ s = \sqrt{a + b\sqrt{2}}$, where $ a$ and $ b$ are positive integers.  What is $ a + b$?

<image1>

Answer: 127

metric geometry - length

Question:
Isosceles $ \triangle ABC$ has a right angle at $ C$.  Point $ P$ is inside $ \triangle ABC$, such that $ PA = 11, PB = 7,$ and $ PC = 6$.  Legs $ \overline{AC}$ and $ \overline{BC}$ have length $ s = \sqrt{a + b\sqrt{2}}$, where $ a$ and $ b$ are positive integers.  What is $ a + b$?

<image1>

Answer: 127


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

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

topology

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


Question:
Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$.  Find $PC$.

<image1>

Answer: 33

metric geometry - length

Question:
Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$.  Find $PC$.

<image1>

Answer: 33


Question: Luis wants to make a pattern by colouring the sides of the triangles shown in the diagram. He wants each triangle to have one red side, one green side and one blue side. Luis has already coloured some of the sides as shown. What colour can he use for the side marked $x$ ? <image1>

Choices:
- only green
- only blue
- only red
- either blue or red
- The task is impossible

Solution: ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-195.jpg?height=217&width=503&top_left_y=831&top_left_x=845)  Label the internal sides of the diagram $a, b, c, d$ and $e$ as shown. The side labelled $a$ is in a triangle with a green side and in a triangle with a blue side and so is to be coloured red. This is also the case for the side labelled $e$. Hence, the side labelled $b$ is in a triangle with a red side and a green side and so is to be coloured blue while the side labelled $d$ is in a triangle with a red side and a blue side and so is to be coloured green. Finally, the side labelled $c$ is in a triangle with a green side and in a triangle with a blue side and so is to be coloured red. Hence the side labelled $x$ is in a triangle with a side that is to be coloured blue (side $b$ ) and with a side that is to be coloured red (side $c$ ). Therefore the side labelled $x$ is to be coloured green.

logic

Question:
Luis wants to make a pattern by colouring the sides of the triangles shown in the diagram. He wants each triangle to have one red side, one green side and one blue side. Luis has already coloured some of the sides as shown. What colour can he use for the side marked $x$ ? <image1>
(A) only green
(B) only blue
(C) only red
(D) either blue or red
(E) The task is impossible


Question:
A $5 \times 5$ square is divided into 25 cells. Initially all its cells are white, as shown. Neighbouring cells are those that share a common edge. On each move two neighbouring cells have their colours changed to the opposite colour (white cells become black and black ones become white). <image1>
What is the minimum number of moves required in order to obtain the chess-like colouring shown on the right?

Answer: 12

Solution: Note that, for each black square that we wish to produce, there will need to be a move which makes it black. This move will not change the colour of any of the other squares which we wish to make black (since the desired black cells are not neighbouring). Since there are 12 such squares, we must necessarily make at least 12 moves.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-222.jpg?height=268&width=274&top_left_y=2176&top_left_x=1462)  However, we can show that 12 moves are sufficient. Consider a pair of black cells with a white cell between them. This colouring can be made in two moves as follows: Starting with WWW, change the colours of two adjacent cells to obtain $\mathrm{BBW}$, then change the middle cell and the one on its right to obtain BWB. By pairing off the 12 black cells into 6 pairs as shown, it is possible to create 12 black cells in $6 \times 2=12$ moves.

combinatorics

Question:
A $5 \times 5$ square is divided into 25 cells. Initially all its cells are white, as shown. Neighbouring cells are those that share a common edge. On each move two neighbouring cells have their colours changed to the opposite colour (white cells become black and black ones become white). <image1>
What is the minimum number of moves required in order to obtain the chess-like colouring shown on the right?

Answer: 12


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

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

metric geometry - angle

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


Question:
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?

<image1>

Answer: 750

metric geometry - area

Question:
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?

<image1>

Answer: 750


Question:
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
<image1>

Answer: 20201

algebra

Question:
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
<image1>

Answer: 20201


Question:
There are six identical circles in the picture. The circles touch the sides of a large rectangle and one another as well. The vertices of the small rectangle lie in the centres of the four circles, as illustrated. The perimeter of the small rectangle is $60 \mathrm{~cm}$. What is the perimeter of the large rectangle?
<image1>

Choices:
- $160 \mathrm{~cm}$
- $140 \mathrm{~cm}$
- $120 \mathrm{~cm}$
- $100 \mathrm{~cm}$
- $80 \mathrm{~cm}$

metric geometry - length

Question:
There are six identical circles in the picture. The circles touch the sides of a large rectangle and one another as well. The vertices of the small rectangle lie in the centres of the four circles, as illustrated. The perimeter of the small rectangle is $60 \mathrm{~cm}$. What is the perimeter of the large rectangle?
<image1>
(A) $160 \mathrm{~cm}$
(B) $140 \mathrm{~cm}$
(C) $120 \mathrm{~cm}$
(D) $100 \mathrm{~cm}$
(E) $80 \mathrm{~cm}$


Question:
<image1>

Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is

Choices:
- 5:12
- 5:13
- 5:19
- 1:4
- 5:21

metric geometry - length

Question:
<image1>

Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is
(A) 5:12
(B) 5:13
(C) 5:19
(D) 1:4
(E) 5:21


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:
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad.  Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles.  The width of the railroad track can be ignored.  How many square miles are in the plot of land ACD?
<image1>

Answer: 4.5

metric geometry - area

Question:
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad.  Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles.  The width of the railroad track can be ignored.  How many square miles are in the plot of land ACD?
<image1>

Answer: 4.5


Question:
An isosceles triangle $P Q R$, in which $P Q=P R$, is split into three separate isosceles triangles, as shown, so that $P S=S Q, R T=R S$ and $Q T=R T$.
What is the size, in degrees, of angle $Q P R$ ? <image1>

Answer: 36

Solution: Let the size, in degrees, of angle $Q P R$ be $x$. Since triangle $P S Q$ is isosceles, angle $P Q S=x$ and, using the external angle theorem, angle $R S T=2 x$. Since triangle $S T R$ is isosceles, angle $S T R=2 x$ and, since angles on a straight line add to $180^{\circ}$, angle $Q T R=180-2 x$. Since triangle $Q T R$ is isosceles and angles in a triangle add to $180^{\circ}$, angle $T Q R=(180-(180-2 x)) / 2=x$. Therefore angle $P Q R=x+x=2 x$ and, since triangle $P Q R$ is also isosceles, angle $P R Q=2 x$. Therefore, in triangle $P Q R$, we have $x+2 x+2 x=180$, since angles in a triangle add to $180^{\circ}$. Hence $x=36$ and so the size, in degrees, of angle $Q P R$ is 36 .

metric geometry - angle

Question:
An isosceles triangle $P Q R$, in which $P Q=P R$, is split into three separate isosceles triangles, as shown, so that $P S=S Q, R T=R S$ and $Q T=R T$.
What is the size, in degrees, of angle $Q P R$ ? <image1>

Answer: 36


Question:
George builds the sculpture shown from seven cubes each of edge length 1. How many more of these cubes must he add to the sculpture so that he builds a large cube of edge length 3?
<image1>

Answer: 20

solid geometry

Question:
George builds the sculpture shown from seven cubes each of edge length 1. How many more of these cubes must he add to the sculpture so that he builds a large cube of edge length 3?
<image1>

Answer: 20


Question:
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
<image1><image2>

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

transformation geometry

Question:
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
<image1><image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Three people walked through the snow in their winter boots. In which order did they walk through the snow?
<image1>
<image2>

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

descriptive geometry

Question:
Three people walked through the snow in their winter boots. In which order did they walk through the snow?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$.  What is the area of $ABCD$?

<image1>

Choices:
- $3\sqrt{5}$
- $10$
- $6\sqrt{5}$
- $20$
- $25$

metric geometry - area

Question:
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$.  What is the area of $ABCD$?

<image1>
(A) $3\sqrt{5}$
(B) $10$
(C) $6\sqrt{5}$
(D) $20$
(E) $25$


Question:
Ten matches are used to make this fish-shaped figure. The piece of string is placed on the shape as shown. The area of the whole shape is 24. What is the area of the shaded triangle?
<image1>

Answer: 2

metric geometry - area

Question:
Ten matches are used to make this fish-shaped figure. The piece of string is placed on the shape as shown. The area of the whole shape is 24. What is the area of the shaded triangle?
<image1>

Answer: 2


Question:
Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown.
The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes
<image1>
underneath it. What is the greatest possible integer that she can write on the top cube?

Answer: 118

Solution: Let the integers written on the small cubes in the bottom layer be arranged as shown.  \begin{tabular}{|l|l|l|} \hline$a$ & $b$ & $c$ \\ \hline$d$ & $e$ & $f$ \\ \hline$g$ & $h$ & $i$ \\ \hline \end{tabular}  Hence, since the integers written on the cubes in the second and third layers are the sum of the integers on the four cubes underneath, the following is written on the cubes in the second layer.  $$ \begin{array}{|l|l|} \hline a+b+d+e & b+c+e+f \\ \hline d+e+g+h & e+f+h+i \\ \hline \end{array} $$  Therefore the integer written on the top cube is  $$ \begin{aligned} & (a+b+d+e)+(b+c+e+f)+(d+e+g+h)+(e+f+h+i) \\ & \quad=(a+b+c+d+e+f+g+h+i)+(b+d+f+h)+3 e \end{aligned} $$  Since the sum of the integers on the bottom layer is 50 , the integer written on the top cube is equal to $50+(b+d+f+h)+3 e$. To maximise this, we first require $e$ to be as large as possible which will be obtained when the other eight integers are as small as possible. Therefore $e=50-(1+2+3+4+5+6+7+8)=14$.  Secondly, $(b+d+f+h)$ should now be made as large as possible and hence $b, d, f$ and $h$ are 5, 6, 7 and 8 in any order. Therefore $(b+d+f+h)=5+6+7+8=26$. Hence the greatest possible integer she can write on the top cube is $50+26+3 \times 14=118$.

algebra

Question:
Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown.
The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes
<image1>
underneath it. What is the greatest possible integer that she can write on the top cube?

Answer: 118


Question:
Segments $O A, O B, O C$ and $O D$ are drawn from the centre $O$ of the square $K L M N$ to its sides so that $O A \perp O B$ and $O C \perp O D$ (as shown in the figure). If the side of the square equals 2, the area of the shaded region equals
<image1>

Answer: 2

metric geometry - area

Question:
Segments $O A, O B, O C$ and $O D$ are drawn from the centre $O$ of the square $K L M N$ to its sides so that $O A \perp O B$ and $O C \perp O D$ (as shown in the figure). If the side of the square equals 2, the area of the shaded region equals
<image1>

Answer: 2


Question:
What is the length of line $A B$ if the side of each of the four squares shown is 1?
<image1>

Choices:
- 5
- $\sqrt{13}$
- $\sqrt{5}+\sqrt{2}$
- $\sqrt{5}$
- None of the previous

metric geometry - length

Question:
What is the length of line $A B$ if the side of each of the four squares shown is 1?
<image1>
(A) 5
(B) $\sqrt{13}$
(C) $\sqrt{5}+\sqrt{2}$
(D) $\sqrt{5}$
(E) None of the previous


Question: The diagram shows a shape made from a regular hexagon of side one unit, six triangles and six squares. What is the perimeter of the shape? <image1>

Choices:
- $6(1+\sqrt{2})$
- $6\left(1+\frac{1}{2} \sqrt{3}\right)$
- $12$
- $6+3 \sqrt{2}$
- $9$

Solution: The interior angles of a regular hexagon are all $120^{\circ}$. At any vertex of the hexagon, there are two squares and a triangle, so the angle of the triangle at that point must be $360^{\circ}-120^{\circ}-90^{\circ}-90^{\circ}=60^{\circ}$. Hence the other two angles of the triangle must add to $180^{\circ}-60^{\circ}=120^{\circ}$. The triangles are isosceles since they have two edges of length 1 unit, so the other two angles are equal and must be $60^{\circ}$. Therefore the triangles are equilateral.  Since the component shapes are all regular, every edge is 1 unit. The perimeter is then 12 units long.

metric geometry - length

Question:
The diagram shows a shape made from a regular hexagon of side one unit, six triangles and six squares. What is the perimeter of the shape? <image1>
(A) $6(1+\sqrt{2})$
(B) $6\left(1+\frac{1}{2} \sqrt{3}\right)$
(C) $12$
(D) $6+3 \sqrt{2}$
(E) $9$


Question:
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
<image1>

Answer: 540

combinatorics

Question:
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
<image1>

Answer: 540


Question:
Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$?
<image1>

Choices:
- $3$
- $\frac{16}{5}$
- $\frac{13}{4}$
- $2\sqrt{3}$
- $\frac{7}{2}$

metric geometry - length

Question:
Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$?
<image1>
(A) $3$
(B) $\frac{16}{5}$
(C) $\frac{13}{4}$
(D) $2\sqrt{3}$
(E) $\frac{7}{2}$


Question:
A right-angled triangle with side lengths $a=8, b=15$ and $c=17$ is given. How big is the radius $r$ of the inscribed semicircle shown?
<image1>

Answer: 4.8

metric geometry - length

Question:
A right-angled triangle with side lengths $a=8, b=15$ and $c=17$ is given. How big is the radius $r$ of the inscribed semicircle shown?
<image1>

Answer: 4.8


Question:
In the diagram drawn on the square grid, find the ratio of the unshaded area to the shaded area.
<image1>

Choices:
- $\frac{1}{4}$
- $\frac{1}{5}$
- $\frac{1}{6}$
- $\frac{2}{5}$
- $\frac{2}{7}$

combinatorial geometry

Question:
In the diagram drawn on the square grid, find the ratio of the unshaded area to the shaded area.
<image1>
(A) $\frac{1}{4}$
(B) $\frac{1}{5}$
(C) $\frac{1}{6}$
(D) $\frac{2}{5}$
(E) $\frac{2}{7}$


Question:
A coin with diameter $1 \mathrm{~cm}$ rolls around the contour outside of a regular hexagon with sides $1 \mathrm{~cm}$ long, as shown. How long is the path traced by the centre of the coin (in $\mathrm{cm}$ )?
<image1>

Choices:
- $6+\frac{\pi}{2}$
- $6+\pi$
- $12+\pi$
- $6+2 \pi$
- $12+2 \pi$

transformation geometry

Question:
A coin with diameter $1 \mathrm{~cm}$ rolls around the contour outside of a regular hexagon with sides $1 \mathrm{~cm}$ long, as shown. How long is the path traced by the centre of the coin (in $\mathrm{cm}$ )?
<image1>
(A) $6+\frac{\pi}{2}$
(B) $6+\pi$
(C) $12+\pi$
(D) $6+2 \pi$
(E) $12+2 \pi$


Question:
Each of the twenty dots on the graph below represents one of Sarah's classmates.  Classmates who are friends are connected with a line segment.  For her birthday party, Sarah is inviting only the following:  all of her friends and all of those classmates who are friends with at least one of her friends.  How many classmates will not be invited to Sarah's party?
<image1>

Answer: 6

graph theory

Question:
Each of the twenty dots on the graph below represents one of Sarah's classmates.  Classmates who are friends are connected with a line segment.  For her birthday party, Sarah is inviting only the following:  all of her friends and all of those classmates who are friends with at least one of her friends.  How many classmates will not be invited to Sarah's party?
<image1>

Answer: 6


Question:
Emily celebrated her birthday on Thursday, and her sister Liepa 8 days earlier. Which weekday was that?
<image1>

Choices:
- Wednesday
- Thursday
- Friday
- Tuesday
- Sunday

arithmetic

Question:
Emily celebrated her birthday on Thursday, and her sister Liepa 8 days earlier. Which weekday was that?
<image1>
(A) Wednesday
(B) Thursday
(C) Friday
(D) Tuesday
(E) Sunday


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

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

logic

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


Question:
We want to paint each square in the grid with the colours P, Q, R and S, so that neighbouring squares always have different colours. (Squares which share the same corner point also count as neighbouring.) Some of the squares are already painted. In which colour(s) could the grey square be painted?
<image1>

Choices:
- only Q
- only R
- only S
- either R or S
- it is not possible.

graph theory

Question:
We want to paint each square in the grid with the colours P, Q, R and S, so that neighbouring squares always have different colours. (Squares which share the same corner point also count as neighbouring.) Some of the squares are already painted. In which colour(s) could the grey square be painted?
<image1>
(A) only Q
(B) only R
(C) only S
(D) either R or S
(E) it is not possible.


Question:
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$?
<image1>

Answer: 100

metric geometry - angle

Question:
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$?
<image1>

Answer: 100


Question:
The teacher wrote the numbers 1 to 8 on the board. Then he covered the numbers with triangles, squares and a circle. The sum of the numbers covered with the triangles equals the sum of the numbers covered with the squares and the number covered with the circle is a quarter of that sum. What is the sum of the numbers covered with the triangles and the circle?
<image1>

Answer: 20

algebra

Question:
The teacher wrote the numbers 1 to 8 on the board. Then he covered the numbers with triangles, squares and a circle. The sum of the numbers covered with the triangles equals the sum of the numbers covered with the squares and the number covered with the circle is a quarter of that sum. What is the sum of the numbers covered with the triangles and the circle?
<image1>

Answer: 20


Question:
Ella wants to write a number into each cell of a $3 \times 3$ grid so that the sum of the numbers in any two cells that share an edge is the same. She has already written two numbers, as shown in the diagram.
When Ella has completed the grid, what will be the sum of all the
<image1>
numbers in the grid?

Answer: 22

Solution: Label the numbers Ella writes down as shown in the diagram.  \begin{tabular}{|c|c|c|} \hline 2 & $a$ & $b$ \\ \hline$c$ & $d$ & 3 \\ \hline$e$ & $f$ & $g$ \\ \hline \end{tabular}  Since the sum of the numbers in any two adjacent cells is the same, $2+a=a+b$ and hence $b=2$. Therefore $b+3=2+3=5$. Hence the sum of the numbers in any two adjacent cells is 5 . It is now straightforward to see that $a=c=f=3$ and that $b=d=e=g=2$. Therefore the sum of all the numbers in the grid is $5 \times 2+4 \times 3=22$.

logic

Question:
Ella wants to write a number into each cell of a $3 \times 3$ grid so that the sum of the numbers in any two cells that share an edge is the same. She has already written two numbers, as shown in the diagram.
When Ella has completed the grid, what will be the sum of all the
<image1>
numbers in the grid?

Answer: 22


Question:
The vertices of a triangle are the points of intersection of the line $y = -x-1$, the line $x=2$, and $y = \frac{1}{5}x+\frac{13}{5}$. Find an equation of the circle passing through all three vertices.

<image1>

Answer: 13

Solution:
Let $A, B,$ and $C$ be the points of intersection in quadrants 4, 1, and 2 respectively. To find the coordinates of $A, B,$ and $C$, we take two line equations at a time and solve for $x$ and $y$. Doing so yields points $A=(2,-3)$, $B=(2,3)$, and $C=(-3,2)$ as vertices of the triangle.

The circle that passes through the three vertices is the circumcircle of the triangle, and by definition, its center is the intersection of the perpendicular bisectors of the triangle's sides. To find the center, it suffices to find two perpendicular bisectors (since the third must pass through the intersection of the first two). We find that the perpendicular bisector of $AB$ is the line $y=0$ and the perpendicular bisector of $AC$ is the line $y=x$. These two perpendicular bisectors intersect at $(0,0)$, which is the center of our desired circle.

To find the radius of our circle, we calculate the distance between the origin and any one of the vertices. The radius has length $\sqrt{13}$. Thus, our circle has formula $(x-0)^2 + (y-0)^2 = (\sqrt{13})^2$, or $\boxed{x^2 + y^2 = 13}$.

analytic geometry

Question:
The vertices of a triangle are the points of intersection of the line $y = -x-1$, the line $x=2$, and $y = \frac{1}{5}x+\frac{13}{5}$. Find an equation of the circle passing through all three vertices.

<image1>

Answer: 13


Question:
Which of the following symbols for signs of the Zodiac has an axis of symmetry?
<image1>

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

transformation geometry

Question:
Which of the following symbols for signs of the Zodiac has an axis of symmetry?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


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

Answer: 500

arithmetic

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

Answer: 500


Question:
Anne has a few grey tiles like the one in the picture.
<image1>
What is the maximum number of these tiles that she can place on the $5 \times 4$ rectangle without any overlaps?
<image2>

Answer: 4

combinatorial geometry

Question:
Anne has a few grey tiles like the one in the picture.
<image1>
What is the maximum number of these tiles that she can place on the $5 \times 4$ rectangle without any overlaps?
<image2>

Answer: 4


Question:
At a pirate school, each student had to sew a black and white flag. The condition was, that the black colour had to cover exactly three fifths of the flag. How many of the following flags fulfilled this condition?
<image1>

Choices:
- None
- One
- Two
- Three
- Four

combinatorial geometry

Question:
At a pirate school, each student had to sew a black and white flag. The condition was, that the black colour had to cover exactly three fifths of the flag. How many of the following flags fulfilled this condition?
<image1>
(A) None
(B) One
(C) Two
(D) Three
(E) Four


Question: Lancelot has drawn a closed path on a cuboid and unfolded it into a net. Which of the nets shown could not be the net of Lancelot's cuboid? <image1>

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

Solution: The nets given can be folded into a solid in only one way. We can continue the path along the edges that would be joined. Connections shown by dashed lines show which pieces are glued together (i.e. belong to to the same path).  ![](https://cdn.mathpix.com/cropped/2023_12_27_6db5618fc6dc2044925bg-10.jpg?height=280&width=1800&top_left_y=2334&top_left_x=240)   Following the path one can see that only $\mathrm{C}$ is not closed.

solid geometry

Question:
Lancelot has drawn a closed path on a cuboid and unfolded it into a net. Which of the nets shown could not be the net of Lancelot's cuboid? <image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: In $\triangle ABC$, what is the value of $x + y$? <image1>

Answer: 90

Solution:
Since $\triangle BDA$ is isosceles, $\angle BAD = \angle ABD = x^\circ$.
Since $\triangle CDA$ is isosceles, $\angle CAD = \angle ACD = y^\circ$. [asy]
import olympiad;
size(7cm);

pair a = dir(76);
pair b = (-1, 0);
pair c = (1, 0);
pair o = (0, 0);

draw(a--b--c--cycle);
draw(a--o);

label("$A$", a, N); label("$B$", b, SW); label("$C$", c, SE); label("$D$", o, S);
label("$104^\circ$", o, 1.8 * NW + 0.4 * E);
label("$x^\circ$", b, 3 * E + NE + NE);
label("$y^\circ$", c, 2 * W + 2 * NW);

add(pathticks(b--o, s=3));
add(pathticks(c--o, s=3));
add(pathticks(a--o, s=3));

label("$x^\circ$", a, 3 * S + 2 * SW + W); label("$y^\circ$", a, 3 * S + SE);

[/asy] Therefore, $\angle BAC = (x + y)^\circ$.

Since the sum of the angles in $\triangle ABC$ is $180^\circ$, we have \begin{align*}
x + y + (x + y) &= 180\\
2x + 2y &= 180\\
x + y &= 90.
\end{align*}Therefore, $x + y = \boxed{90}$.

metric geometry - angle

Question: In $\triangle ABC$, what is the value of $x + y$? <image1>

Answer: 90


Question:
The figure shows a shape consisting of a regular hexagon of side $18 \mathrm{~cm}$, six triangles and six squares. The outer perimeter of the shape is $P \mathrm{~cm}$. What is the value of $P$ ?
<image1>

Answer: 216

Solution: Each of the squares has a side-length of $18 \mathrm{~cm}$. Therefore each of the triangles has two sides of length $18 \mathrm{~cm}$. Hence the triangles are isosceles. Let the angle contained by the two $18 \mathrm{~cm}$ sides of these triangle be $x^{\circ}$. The interior angles of a square and a regular hexagon are $90^{\circ}$ and $120^{\circ}$ respectively. By considering angles at a point we have $x+90+90+120=360$. Therefore $x=60$ and the triangles are equilateral. All twelve outer edges of the figure are $18 \mathrm{~cm}$ in length. Therefore $P=12 \times 18=216$.

metric geometry - length

Question:
The figure shows a shape consisting of a regular hexagon of side $18 \mathrm{~cm}$, six triangles and six squares. The outer perimeter of the shape is $P \mathrm{~cm}$. What is the value of $P$ ?
<image1>

Answer: 216


Question:
A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.
<image1>

The inequalities
\[ |x| + |y| \leq \sqrt{2(x^2 + y^2)} \leq 2\mbox{Max}(|x|, |y|)\]
are represented geometrically* by the figure numbered

Choices:
- $I$
- $II$
- $III$
- $IV$
- $\mbox{none of these}*An inequality of the form  f(x, y) \leq g(x, y), for all  x and  y is represented geometrically by a figure showing the containment \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}for a typical real number  a$.

analytic geometry

Question:
A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.
<image1>

The inequalities
\[ |x| + |y| \leq \sqrt{2(x^2 + y^2)} \leq 2\mbox{Max}(|x|, |y|)\]
are represented geometrically* by the figure numbered
(A) $I$
(B) $II$
(C) $III$
(D) $IV$
(E) $\mbox{none of these}*An inequality of the form  f(x, y) \leq g(x, y), for all  x and  y is represented geometrically by a figure showing the containment \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}for a typical real number  a$.


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

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

analytic geometry

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


Question: The diagram on the right shows nine points in a square array. What is the smallest number of points that need to be removed in order that no three of the remaining points are in a straight line? <image1>

Answer: 3

Solution: At least one point must be removed from each of the three horizontal lines; so at least three points need to be removed. However, removing the three points lying on either diagonal does what is required.

combinatorial geometry

Question: The diagram on the right shows nine points in a square array. What is the smallest number of points that need to be removed in order that no three of the remaining points are in a straight line? <image1>

Answer: 3


Question:
The distance between two shelves in Monika's kitchen is $36 \mathrm{~cm}$. She knows that a stack of 8 identical glasses is $42 \mathrm{~cm}$ high and a stack of 2 such glasses is $18 \mathrm{~cm}$ high. How many glasses has the biggest stack that will fit between two shelves?
<image1>

Answer: 6

algebra

Question:
The distance between two shelves in Monika's kitchen is $36 \mathrm{~cm}$. She knows that a stack of 8 identical glasses is $42 \mathrm{~cm}$ high and a stack of 2 such glasses is $18 \mathrm{~cm}$ high. How many glasses has the biggest stack that will fit between two shelves?
<image1>

Answer: 6


Question: In the diagram on the right, the number in each circle is the sum of the numbers in the two circles below it. What is the value of $x$ ? <image1>

Answer: 82

Solution: The empty spaces in the diagram can be completed as shown. Hence the value of $x$ is $35+47=82$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_bfd7d7e47aafe1ef66ecg-1.jpg?height=360&width=485&top_left_y=2233&top_left_x=1345)

algebra

Question: In the diagram on the right, the number in each circle is the sum of the numbers in the two circles below it. What is the value of $x$ ? <image1>

Answer: 82


Question:
Six figures were drawn, one on each side of a cube, as shown beside, in different positions. On the side that does not appear beside is this drawing:
<image1>
What is the figure on the face opposite to it?
<image2>
<image3>

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

descriptive geometry

Question:
Six figures were drawn, one on each side of a cube, as shown beside, in different positions. On the side that does not appear beside is this drawing:
<image1>
What is the figure on the face opposite to it?
<image2>
<image3>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: Matchsticks can be used to write digits, as shown in the diagram. How many different positive integers can be written using exactly six matchsticks in this way? <image1>

Answer: 6

Solution: The number of matchsticks used for each of the digits 0 to 9 is $6,2,5,5,4,5,6,3,7$ and 6 respectively. Hence, the positive integers that can be made using exactly six matches are 6, 9, 14, 41, 77 and 1Therefore 6 different positive integers can be made.

combinatorics

Question: Matchsticks can be used to write digits, as shown in the diagram. How many different positive integers can be written using exactly six matchsticks in this way? <image1>

Answer: 6


Question:
<image1>

If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is

Answer: 18

metric geometry - area

Question:
<image1>

If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is

Answer: 18


Question:
A rectangle with perimeter $30 \mathrm{~cm}$ is divided by two lines, forming a square of area $9 \mathrm{~cm}^{2}$, as shown in the figure.
<image1>
What is the perimeter of the shaded rectangle?

Choices:
- $14 \mathrm{~cm}$
- $16 \mathrm{~cm}$
- $18 \mathrm{~cm}$
- $21 \mathrm{~cm}$
- $24 \mathrm{~cm}$

Solution: The square has sides of length $\sqrt{9} \mathrm{~cm}=3 \mathrm{~cm}$. Let $x$ and $y$ be the width and length in $\mathrm{cm}$ of the shaded rectangle. Then the large rectangle has perimeter $2(3+x+3+y)=30$. This gives $6+x+y=15$ so $x+y=9$. Hence the perimeter in $\mathrm{cm}$ of the shaded rectangle is $2(x+y)=2 \times 9=18$.

metric geometry - length

Question:
A rectangle with perimeter $30 \mathrm{~cm}$ is divided by two lines, forming a square of area $9 \mathrm{~cm}^{2}$, as shown in the figure.
<image1>
What is the perimeter of the shaded rectangle?
(A) $14 \mathrm{~cm}$
(B) $16 \mathrm{~cm}$
(C) $18 \mathrm{~cm}$
(D) $21 \mathrm{~cm}$
(E) $24 \mathrm{~cm}$


Question:
The diagram <image1> shows the number 8. A dot stands for the number 1 and a line for the number 5. Which diagram represents the number 12?
<image2>

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

arithmetic

Question:
The diagram <image1> shows the number 8. A dot stands for the number 1 and a line for the number 5. Which diagram represents the number 12?
<image2>
(A) (A)
(B) (B)
(C) (C)
(D) (D)
(E) (E)


Question:
Let $A B C$ be a triangle with area 30. Let $D$ be any point in its interior and let $e, f$ and $g$ denote the distances from $D$ to the sides of the triangle. What is the value of the expression $5 e+12 f+13 g$?
<image1>

Answer: 60

metric geometry - length

Question:
Let $A B C$ be a triangle with area 30. Let $D$ be any point in its interior and let $e, f$ and $g$ denote the distances from $D$ to the sides of the triangle. What is the value of the expression $5 e+12 f+13 g$?
<image1>

Answer: 60


Question: We define the $\emph{weight}$ of a path to be the sum of the numbers written on each edge of the path. Find the minimum weight among all paths in the graph below that visit each vertex precisely once. \n<image1>

Answer: 65

Solution: Very easy casework shows that up to rotation and reflection, there are three types of paths in\n        total, of the following forms: Among all paths of the first kind, the one with the minimal weight has weight\n        16 + 11 + 17 + 14 + 3 + 19 + 3 + 9 + min{4 - max{16, 11}, 2 - max{17, 14}, . . . }\n        which is 77 by inspection. Among all paths of the second kind, the one with the minimal weight has weight\nmin{9 + 3 + 19 + 3 + 4 + 2 + min{16 + 17, 11 + 14}, 3 + 9 + 16 + 11 + 8 + 2 + min{19 + 14, 3 + 17}, . . . }\nwhich is 65 by inspection. Among all paths of the third kind, the one with the minimal weight has weight\nmin{3 + 9 + 8 + 4 + 14 + 17 + min{3 + 16, 19 + 11}, 16 + 11 + 4 + 2 + 3 + 19 + min{9 + 14, 3 + 17}}\nwhich is 74 by inspection. Thus, the minimum weight spanning path has weight 65 .

graph theory

Question: We define the $\emph{weight}$ of a path to be the sum of the numbers written on each edge of the path. Find the minimum weight among all paths in the graph below that visit each vertex precisely once. \n<image1>

Answer: 65