Question: A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system:\n\n$\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$.\n\n$\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don't know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values.\n\n$\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal.\n\nThey then provide you with the following equations, written in Gliesian, which are known to be true:<image1>\n What is the human number equivalent of $๑$ ?

Answer: $\frac{1}{3}$

Solution: A natural first step would be to try and determine whether $\gamma=+$ or $\eta=+$. In terms of properties, the one thing that distinguishes addition from multiplication is the distributive property; that is, $(a+b) \cdot c=a c+b c$, but $(a \cdot b)+c$ may not equal $(a+c) \cdot(b+c)$. Using the distributive property would require a set of three equations using the same operation, and sharing at least one character. Two characters fit that requirement; $>$ with $\gamma$, and $\mathbb{U}$ with $\eta$.\nLet's focus on $\amalg$. We see that the three values $\eta$ 'd into $\uplus$ are $\gtrdot, \pitchfork$, and $\square$. Since those would correspond to the $a, b$, and $a+b$ in the distributive property, we should search for an equation with $\square, \gtrdot, \pitchfork$, and $\gamma$. There is one of those, fortunately; that one says that $\square \gamma \gtrdot=\pitchfork$. If we assume $\eta$ is multiplication and $\gamma$ is addition, then that yields the statement $(\square+\gtrdot) \cdot ய=\pitchfork \cdot ய=\square$, and $\square \cdot(\uplus)+\gtrdot \cdot \cup)>+\diamond=\triangleright$. This would imply that $>=\diamond$. Therefore, $\eta$ cannot equal multiplication, so $\eta=+$ and $\gamma=\cdot$.\n\n    If we were to focus on $>$, then we could try something similar; the three values $\gamma$ 'd into $>$ are $\diamond, \square$, and $\ltimes$. Unfortunately, no equation uses all three of those symbols, so we can't get any information from that process.\nNow that we know that $\eta=+$ and $\gamma=\cdot$, let's try to solve this problem. Since we are looking for (๑), we should focus on the equation using (๑); the only one using that is $\odot \cdot \varkappa=>$. Let's try substituting this as far as we can, until only (๑) and $>$ remain. Note that $>$ can't equal zero, since it is multiplied into numbers to yield results not equal to $\gg$ :\n        (ㅇ) $\cdot x=\gg$\n(ㅇ) $\cdot(>+x)=>$\n        (ㅇ) $\frac{>+\lambda}{>}=>$, since $>$ can't be 0\n        () $\cdot\left(>+\frac{\pitchfork+b}{>}\right)=>$\n() $\cdot\left(>+\frac{\square \cdot \gtrdot+\gtrdot \cdot \Delta}{\gtrdot}\right)=>$\n        (๑) $\cdot(\gtrdot+(\square+\diamond))=>$\n        (๑) $\cdot(>+(\square+>+U))=>$ Note: This is the only step where multiple equations can be used.\n        (๑) $\cdot(>+(>+>))=>$\n        (6) $\cdot(3 \cdot \gtrdot)=>$\n        (ㅇ) $=\frac{1}{3}$ as desired.

algebra

Question: A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system:\n\n$\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$.\n\n$\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don't know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values.\n\n$\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal.\n\nThey then provide you with the following equations, written in Gliesian, which are known to be true:<image1>\n What is the human number equivalent of $๑$ ?

Answer: $\frac{1}{3}$


Question:
In the box are seven blocks. It is possible to slide the blocks around so that another block can be added to the box. What is the minimum number of blocks that must be moved?
<image1>

Answer: 3

transformation geometry

Question:
In the box are seven blocks. It is possible to slide the blocks around so that another block can be added to the box. What is the minimum number of blocks that must be moved?
<image1>

Answer: 3


Question:
Eva has a pair a scissors and five letters made from cardboard. She cuts up each letter with a single straight cut so that as many pieces as possible are obtained. For which letter does she obtain the most pieces?
<image1>

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

transformation geometry

Question:
Eva has a pair a scissors and five letters made from cardboard. She cuts up each letter with a single straight cut so that as many pieces as possible are obtained. For which letter does she obtain the most pieces?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is $56$. The area of the region bounded by the polygon is
<image1>

Answer: 100

metric geometry - area

Question:
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is $56$. The area of the region bounded by the polygon is
<image1>

Answer: 100


Question:
The distance of two not-crossing edges of a regular tetrahedron (triangular pyramid with all the six edges equal) is $6 \mathrm{~cm}$. What is the volume of the tetrahedron $\left(\right.$ in $\left.\mathrm{cm}^{3}\right)$?
<image1>

Answer: 72

solid geometry

Question:
The distance of two not-crossing edges of a regular tetrahedron (triangular pyramid with all the six edges equal) is $6 \mathrm{~cm}$. What is the volume of the tetrahedron $\left(\right.$ in $\left.\mathrm{cm}^{3}\right)$?
<image1>

Answer: 72


Question:
Heinzi the kangaroo has bought some toys. For this he gave 150 Kangoo-coins (KC) and received 20 kangoo-coins back. Just before leaving the shop he changed his mind, and exchanged one of the toys he had bought with another one. Therefore he received a further 5 kangoo-coins back from the shopkeeper. Which of the toys in the picture has Heinzi taken home with him?
<image1>

Choices:
- Carriage and Aeroplane
- Carriage and Bus
- Carriage and Tram
- Motorbike and Tram
- Bus, Motorbike and Tram

algebra

Question:
Heinzi the kangaroo has bought some toys. For this he gave 150 Kangoo-coins (KC) and received 20 kangoo-coins back. Just before leaving the shop he changed his mind, and exchanged one of the toys he had bought with another one. Therefore he received a further 5 kangoo-coins back from the shopkeeper. Which of the toys in the picture has Heinzi taken home with him?
<image1>
(A) Carriage and Aeroplane
(B) Carriage and Bus
(C) Carriage and Tram
(D) Motorbike and Tram
(E) Bus, Motorbike and Tram


Question: Sam spends his days walking around the following $2\times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?\n<image1>

Answer: 167

Solution: Firstly, we claim that the shortest possible sequence is $8$ moves long. We observe that it must be strictly longer than $5$ moves because the average move adds less than $4$; if it's $6$ moves, then the total sum is odd because we are adding three odds and three evens; if it's $7$ moves, the total sum is also still odd.\n\nThe longest possible sequence is the sequence $2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2$, which is $13$ moves long.\n\nWe begin by considering $8$-move sequences. Begin with the $8$-tuple $(2,1,2,1,2,1,2,1)$; the current sum is $12$, so we need to add $2$ to four of these elements. This yields $\binom{8}{4} = 70$ possibilities.\n\nNext, $9$-move sequences. $(2,1,2,1,2,1,2,1,2)$ has a sum of $14$, so we need to add $2$ to three of these elements. This yields $\binom{9}{3} = 84$ possibilities.\n\nThere are no $10$ or $11$-move sequences for parity reasons.\n\n$12$-move sequences: $(2,1,2,1,2,1,2,1,2,1,2,1)$ has a sum of $18$, so we add $2$ to one element. This gives $12$ possibilities. Finally, there is exactly one $13$-move sequence.\n\nOur total is $70 + 84 + 12 + 1 = \boxed{167}.$

combinatorics

Question: Sam spends his days walking around the following $2\times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?\n<image1>

Answer: 167


Question: The diagram shows a shape made of arcs of three circles, each with radius $R$. The centres of the circles lie on the same straight line, and the middle circle passes through the centres of the other two circles. What is the perimeter of the shape? <image1>

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

Solution: By drawing radii and chords as shown, we can see that the triangles are equilateral and therefore each of the angles is $60^{\circ}$. Hence the left and right circles have each lost $120^{\circ}$ (one third) of their circumferences, and the central circle has one-third $\left(\frac{1}{6}+\frac{1}{6}\right)$ of its circumference remaining. Since $1-\frac{1}{3}+1-\frac{1}{3}+\frac{1}{3}=\frac{5}{3}$, the perimeter of the shape is  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-267.jpg?height=223&width=417&top_left_y=1356&top_left_x=1553) $\frac{5}{3} \times 2 \pi R=\frac{10 \pi R}{3}$.

metric geometry - length

Question:
The diagram shows a shape made of arcs of three circles, each with radius $R$. The centres of the circles lie on the same straight line, and the middle circle passes through the centres of the other two circles. What is the perimeter of the shape? <image1>
(A) $\frac{2 \pi R \sqrt{3}}{3}$
(B) $\frac{5 \pi R}{3}$
(C) $\frac{10 \pi R}{3}$
(D) $2 \pi R \sqrt{3}$
(E) $4 \pi R$


Question: A belt is drawn tightly around three circles of radius $10$ cm each, as shown. The length of the belt, in cm, can be written in the form $a + b\pi$ for rational numbers $a$ and $b$. What is the value of $a + b$? <image1>

Answer: 80

Solution:
We break the belt into six pieces, three where the belt touches no circle and three where it does.

First consider the portion of the belt that does not touch a circle.  Each segment is the length of two radii, or $20$ cm.  There are three such segments, or $60$ cm in total.

Now consider the portion of the belt that does touch a circle.  Because there are three circles, the belt will touch each circle for $\frac{1}{3}$ of its circumference.  Since it does this three times, this is the length of these segments combined, which is the circumference of a full circle, which is $20\pi$ cm for a circle of radius $10$ cm.

Therefore the length of the belt is $60 + 20\pi$ cm.  From this we conclude that $a = 60$ and $b = 20,$ and so $a+b = \boxed{80}.$

metric geometry - length

Question: A belt is drawn tightly around three circles of radius $10$ cm each, as shown. The length of the belt, in cm, can be written in the form $a + b\pi$ for rational numbers $a$ and $b$. What is the value of $a + b$? <image1>

Answer: 80


Question:
The diagram shows the net of a cube whose faces are numbered. Sascha adds the numbers that are on opposite faces of the cube. Which three results does he get?
<image1>

Choices:
- $4,6,11$
- $4,5,12$
- $5,6,10$
- $5,7,9$
- $5,8,8$

descriptive geometry

Question:
The diagram shows the net of a cube whose faces are numbered. Sascha adds the numbers that are on opposite faces of the cube. Which three results does he get?
<image1>
(A) $4,6,11$
(B) $4,5,12$
(C) $5,6,10$
(D) $5,7,9$
(E) $5,8,8$


Question:
Anna has <image1>.
Barbara gave Eva <image2>.
Josef has a <image3>.
Bob has <image4>.
Who is Barbara?
<image5>

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

logic

Question:
Anna has <image1>.
Barbara gave Eva <image2>.
Josef has a <image3>.
Bob has <image4>.
Who is Barbara?
<image5>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person (not the original number the person picked). The number picked by the person who announced the average $6$ was
<image1>

Choices:
- $1$
- $5$
- $6$
- $10$
- $\text{not uniquely determined from the given information}$

algebra

Question:
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person (not the original number the person picked). The number picked by the person who announced the average $6$ was
<image1>
(A) $1$
(B) $5$
(C) $6$
(D) $10$
(E) $\text{not uniquely determined from the given information}$


Question:
The two markers with a question mark have the same number.
<image1>
Which number do you have to put instead of the question mark so that the calculation is correct?

Answer: 3

algebra

Question:
The two markers with a question mark have the same number.
<image1>
Which number do you have to put instead of the question mark so that the calculation is correct?

Answer: 3


Question: The diagram shows a solid with six triangular faces. At each vertex there is a number and two of the numbers are 1 and 5, as shown. For each face the sum of the numbers at the three vertices of each face is calculated, and all the sums are the same. What is the sum of all five numbers at the vertices? <image1>

Answer: 17

Solution: Let the numbers at two of the other vertices be $u$ and $v$, as shown in the diagram on the right. The three faces sharing the vertex labelled with the number 1 all have the same sum. Then $1+v+u=1+5+u$ and so $v=5$.  Similarly, $1+v+5=1+v+u$ so $u=5$. Hence the sum for each face is $1+5+5$, i.e. 11 , and we see that the number at the bottom vertex is 1 . The total of all the  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-098.jpg?height=294&width=348&top_left_y=504&top_left_x=1482) vertices is $1+5+5+5+1=17$.

solid geometry

Question: The diagram shows a solid with six triangular faces. At each vertex there is a number and two of the numbers are 1 and 5, as shown. For each face the sum of the numbers at the three vertices of each face is calculated, and all the sums are the same. What is the sum of all five numbers at the vertices? <image1>

Answer: 17


Question: Sepideh is making a magic multiplication square using the numbers 1, 2, 4, 5, 10, 20, 25, 50 and 100 . The products of the numbers in each row, in each column and in the two diagonals should all be the same. In the figure you can see how she has started. Which number should Sepideh place in the cell with the question mark? <image1>

Answer: 4

Solution: Since each of the nine numbers appears exactly once in the three rows, the product of each of the three rows is equal to the product of all nine numbers, that is $1 \times 2 \times 4 \times 5 \times 10 \times 20 \times 25 \times 50 \times 100$. This equals $1 \times 2 \times 2^{2} \times 5 \times 2 \times 5 \times 2^{2} \times 5 \times 5^{2} \times 2 \times 5^{2} \times 2^{2} \times 5^{2}=2^{9} \times 5^{9}$. Since the products of each of the three rows are equal, these products are equal to $\sqrt[3]{2^{9} \times 5^{9}}=2^{3} \times 5^{3}$. So the 'magic product' is 1000 .  By considering the top row, we can see that the top right cell must contain $1000 \div(20 \times 1)=50$.  The only other way to get 1000 as the product of 1 and two of the remaining numbers is $1 \times 10 \times 100$. So 10 and 100 must fill the two spaces below 1 . We cannot put the 100 in the centre since the products of the numbers on each diagonal would then be too large. So 10 goes in the centre and 100 below it. From the diagonal we see that the bottom right entry is 5 and so the middle number on the right is 4.  Note: It may interest readers to know that in a $3 \times 3$ multiplicative magic square the centre number is always the cube root of the magic total.

combinatorics

Question: Sepideh is making a magic multiplication square using the numbers 1, 2, 4, 5, 10, 20, 25, 50 and 100 . The products of the numbers in each row, in each column and in the two diagonals should all be the same. In the figure you can see how she has started. Which number should Sepideh place in the cell with the question mark? <image1>

Answer: 4


Question:
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)

<image1>

Choices:
- $\frac{1}{6}$
- $\frac{1}{7}$
- $\frac{1}{8}$
- $\frac{1}{12}$
- $\frac{1}{16}$

metric geometry - area

Question:
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)

<image1>
(A) $\frac{1}{6}$
(B) $\frac{1}{7}$
(C) $\frac{1}{8}$
(D) $\frac{1}{12}$
(E) $\frac{1}{16}$


Question:
Rita wants to write a number into every square of the diagram shown. Every number should be equal to the sum of the two numbers from the adjacent squares. Squares are adjacent if they share one edge. Two numbers are already given. Which number is she going to write into the square marked with $x$?
<image1>

Answer: 7

logic

Question:
Rita wants to write a number into every square of the diagram shown. Every number should be equal to the sum of the two numbers from the adjacent squares. Squares are adjacent if they share one edge. Two numbers are already given. Which number is she going to write into the square marked with $x$?
<image1>

Answer: 7


Question:
In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\frac{1}{n}$, where $n$ is a positive integer. Find $n$.
<image1>

Answer: 429

transformation geometry

Question:
In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\frac{1}{n}$, where $n$ is a positive integer. Find $n$.
<image1>

Answer: 429


Question:
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

<image1>

Choices:
- $27\sqrt{3}-9\pi$
- $27\sqrt{3}-6\pi$
- $54\sqrt{3}-18\pi$
- $54\sqrt{3}-12\pi$
- $108\sqrt{3}-9\pi$

metric geometry - area

Question:
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

<image1>
(A) $27\sqrt{3}-9\pi$
(B) $27\sqrt{3}-6\pi$
(C) $54\sqrt{3}-18\pi$
(D) $54\sqrt{3}-12\pi$
(E) $108\sqrt{3}-9\pi$


Question:
How much does this Ferris wheel need to turn so that a white gondola is on top for
the first time?
<image1>

Choices:
- $\frac{1}{2}$ turn
- $\frac{1}{3}$ turn
- $\frac{1}{6}$ turn
- $\frac{1}{12}$ turn
- $\frac{5}{6}$ turn

arithmetic

Question:
How much does this Ferris wheel need to turn so that a white gondola is on top for
the first time?
<image1>
(A) $\frac{1}{2}$ turn
(B) $\frac{1}{3}$ turn
(C) $\frac{1}{6}$ turn
(D) $\frac{1}{12}$ turn
(E) $\frac{5}{6}$ turn


Question:
Four of the numbers 1,3,4,5 and 7 are written into the boxes so that the calculation is correct.
Which number was not used?
<image1>

Answer: 4

arithmetic

Question:
Four of the numbers 1,3,4,5 and 7 are written into the boxes so that the calculation is correct.
Which number was not used?
<image1>

Answer: 4


Question:
In right triangle $ \triangle ACE$, we have $ AC = 12$, $ CE = 16$, and $ EA = 20$. Points $ B$, $ D$, and $ F$ are located on $ \overline{AC}$, $ \overline{CE}$, and $ \overline{EA}$, respectively, so that $ AB = 3$, $ CD = 4$, and $ EF = 5$. What is the ratio of the area of $ \triangle DBF$ to that of $ \triangle ACE$?
<image1>

Choices:
- $\frac{1}{4}$
- $\frac{9}{25}$
- $\frac{3}{8}$
- $\frac{11}{25}$
- $\frac{7}{16}$

metric geometry - area

Question:
In right triangle $ \triangle ACE$, we have $ AC = 12$, $ CE = 16$, and $ EA = 20$. Points $ B$, $ D$, and $ F$ are located on $ \overline{AC}$, $ \overline{CE}$, and $ \overline{EA}$, respectively, so that $ AB = 3$, $ CD = 4$, and $ EF = 5$. What is the ratio of the area of $ \triangle DBF$ to that of $ \triangle ACE$?
<image1>
(A) $\frac{1}{4}$
(B) $\frac{9}{25}$
(C) $\frac{3}{8}$
(D) $\frac{11}{25}$
(E) $\frac{7}{16}$


Question:
In a list of five numbers the first number is 2 and the last one is 12. The product of the first three numbers is 30 , of the middle three 90 and of the last three 360. What is the middle number in that list?
<image1>

Answer: 5

algebra

Question:
In a list of five numbers the first number is 2 and the last one is 12. The product of the first three numbers is 30 , of the middle three 90 and of the last three 360. What is the middle number in that list?
<image1>

Answer: 5


Question:
Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that appear in the second row from the top?

<image1>

Answer: 367

algebra

Question:
Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that appear in the second row from the top?

<image1>

Answer: 367


Question:
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
<image1>

Answer: 539

metric geometry - area

Question:
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
<image1>

Answer: 539


Question:
A triangle is folded along the dashed line as shown. The area of the triangle is 1.5 times the area of the resulting figure. We know that the total area of the grey parts is 1. Determine the area of the starting triangle.
<image1>

Answer: 3

transformation geometry

Question:
A triangle is folded along the dashed line as shown. The area of the triangle is 1.5 times the area of the resulting figure. We know that the total area of the grey parts is 1. Determine the area of the starting triangle.
<image1>

Answer: 3


Question:
Michael has two building blocks. Each building block is made up of two cubes glued together. Which figure can he not make using the blocks?
<image1>
<image2>

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

combinatorial geometry

Question:
Michael has two building blocks. Each building block is made up of two cubes glued together. Which figure can he not make using the blocks?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Beatriz has five sisters with ages of 2, 3, 5, 8, 10 and 17. Beatriz writes these ages in the circles of the opposite diagram, so that the sum of the ages in the four corners of the square is equal to the sum of the ages in the four circles aligned horizontally. What is this sum?
<image1>

Answer: 32

algebra

Question:
Beatriz has five sisters with ages of 2, 3, 5, 8, 10 and 17. Beatriz writes these ages in the circles of the opposite diagram, so that the sum of the ages in the four corners of the square is equal to the sum of the ages in the four circles aligned horizontally. What is this sum?
<image1>

Answer: 32


Question:
Six coins build a triangle (see picture). What is the smallest number of coins that must be moved to create the circle?
<image1>

Answer: 2

combinatorial geometry

Question:
Six coins build a triangle (see picture). What is the smallest number of coins that must be moved to create the circle?
<image1>

Answer: 2


Question:
The diagram consists of three squares each one of side length 1. The midpoint of the topmost square is exactly above the common side of the two other squares. What is the area of the section coloured grey?
<image1>

Answer: 1

combinatorial geometry

Question:
The diagram consists of three squares each one of side length 1. The midpoint of the topmost square is exactly above the common side of the two other squares. What is the area of the section coloured grey?
<image1>

Answer: 1


Question: The diagram shows a shape made from six squares, each measuring $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$. The shape has perimeter of length $14 \mathrm{~cm}$. The zigzag shape is then continued until it has 2013 squares. What is the length, in $\mathrm{cm}$, of the perimeter of the new shape? <image1>

Answer: 4028

Solution: The two squares at either end of the shape contribute $3 \mathrm{~cm}$ towards the total perimeter of the zigzag. Each of the other 2011 squares contribute $2 \mathrm{~cm}$ towards the perimeter of the overall shape. Thus the perimeter of the zigzag is  $2 \times 3+2011 \times 2=4028 \mathrm{~cm}$.

combinatorial geometry

Question: The diagram shows a shape made from six squares, each measuring $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$. The shape has perimeter of length $14 \mathrm{~cm}$. The zigzag shape is then continued until it has 2013 squares. What is the length, in $\mathrm{cm}$, of the perimeter of the new shape? <image1>

Answer: 4028


Question:
In the diagram, $A B$ has length $1 ; \angle A B C=\angle A C D=90^{\circ}$; $\angle C A B=\angle D A C=\theta$. What is the length of $A D$?
<image1>

Choices:
- $\cos \beta+\tg \beta$
- $\frac{1}{\cos (2 \beta)}$
- $\cos ^{2} \beta$
- $\cos (2 \beta)$
- $\frac{1}{\cos ^{2} \beta}$

metric geometry - length

Question:
In the diagram, $A B$ has length $1 ; \angle A B C=\angle A C D=90^{\circ}$; $\angle C A B=\angle D A C=\theta$. What is the length of $A D$?
<image1>
(A) $\cos \beta+\tg \beta$
(B) $\frac{1}{\cos (2 \beta)}$
(C) $\cos ^{2} \beta$
(D) $\cos (2 \beta)$
(E) $\frac{1}{\cos ^{2} \beta}$


Question:
In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$?
<image1>

Choices:
- $\frac{1}{6}$
- $\frac{2}{9}$
- $\frac{5}{18}$
- $\frac{1}{3}$
- $\frac{7}{20}$

metric geometry - area

Question:
In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$?
<image1>
(A) $\frac{1}{6}$
(B) $\frac{2}{9}$
(C) $\frac{5}{18}$
(D) $\frac{1}{3}$
(E) $\frac{7}{20}$


Question:
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has length

<image1>

Choices:
- $20$
- $15\sqrt{2}$
- $24$
- $25$
- $\text{none of these}$

metric geometry - length

Question:
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has length

<image1>
(A) $20$
(B) $15\sqrt{2}$
(C) $24$
(D) $25$
(E) $\text{none of these}$


Question:
A die is in the position shown in the picture. It can be rolled along the path of 12 squares as shown. How many times must the die go around the path in order for it to return to its initial position with all faces in the initial positions?
<image1>

Answer: 3

solid geometry

Question:
A die is in the position shown in the picture. It can be rolled along the path of 12 squares as shown. How many times must the die go around the path in order for it to return to its initial position with all faces in the initial positions?
<image1>

Answer: 3


Question:
A cube has diagonals drawn on three adjacent faces as shown in the diagram. Which of the following nets could Usman use to make the cube shown?
<image1>
<image2>

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

Solution: On each net, label the four vertices of the right-hand square 1,2,3 and 4 as shown. Also label any vertex on any of the other squares that will meet vertices 1,2,3 or 4 when the net of the cube is assembled into a cube with the corresponding value. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f598bb42a47f3275d94g-2.jpg?height=246&width=1530&top_left_y=536&top_left_x=407)  Since there are three vertices of the original cube at which two diagonals meet, to be a suitable net for the cube shown, any diagonal drawn meets another diagonal at a vertex with the same label. As can be seen, only in net $\mathrm{D}$ are the ends of the diagonals at vertices with the same label. Therefore Usman could only use net $\mathrm{D}$ to make the cube shown.

solid geometry

Question:
A cube has diagonals drawn on three adjacent faces as shown in the diagram. Which of the following nets could Usman use to make the cube shown?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


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

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

combinatorics

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


Question:
How many faces has the object shown? (Prism with a hole)
<image1>

Answer: 8

counting

Question:
How many faces has the object shown? (Prism with a hole)
<image1>

Answer: 8


Question:
A rectangle with length $24 \mathrm{~m}$ and width $1 \mathrm{~m}$ is cut into smaller rectangles, each with width $1 \mathrm{~m}$. There are four pieces with length $4 \mathrm{~m}$, two pieces with length $3 \mathrm{~m}$ and one piece with length $2 \mathrm{~m}$. These smaller rectangles are put together to form another rectangle. What is the smallest possible perimeter of the new rectangle?
<image1>

Choices:
- $14 \mathrm{~m}$
- $20 \mathrm{~m}$
- $22 \mathrm{~m}$
- $25 \mathrm{~m}$
- $28 \mathrm{~m}$

metric geometry - area

Question:
A rectangle with length $24 \mathrm{~m}$ and width $1 \mathrm{~m}$ is cut into smaller rectangles, each with width $1 \mathrm{~m}$. There are four pieces with length $4 \mathrm{~m}$, two pieces with length $3 \mathrm{~m}$ and one piece with length $2 \mathrm{~m}$. These smaller rectangles are put together to form another rectangle. What is the smallest possible perimeter of the new rectangle?
<image1>
(A) $14 \mathrm{~m}$
(B) $20 \mathrm{~m}$
(C) $22 \mathrm{~m}$
(D) $25 \mathrm{~m}$
(E) $28 \mathrm{~m}$


Question:
A rectangular garden was $50 \mathrm{~m}$ long and $40 \mathrm{~m}$ wide. An artificial lake was built next to it, so that the whole set forms a $60 \mathrm{~m}$ square. Then a fence was stretched, separating both the garden and the lake in two parts with equal areas, as shown in the picture. How long is this fence?
<image1>

Choices:
- $60 \mathrm{~m}$
- $30 \sqrt{5} \mathrm{~m}$
- $60 \sqrt{2} \mathrm{~m}$
- $85 \mathrm{~m}$
- $60 \sqrt{3} \mathrm{~m}$

metric geometry - length

Question:
A rectangular garden was $50 \mathrm{~m}$ long and $40 \mathrm{~m}$ wide. An artificial lake was built next to it, so that the whole set forms a $60 \mathrm{~m}$ square. Then a fence was stretched, separating both the garden and the lake in two parts with equal areas, as shown in the picture. How long is this fence?
<image1>
(A) $60 \mathrm{~m}$
(B) $30 \sqrt{5} \mathrm{~m}$
(C) $60 \sqrt{2} \mathrm{~m}$
(D) $85 \mathrm{~m}$
(E) $60 \sqrt{3} \mathrm{~m}$


Question:
In triangle $A B C$ (see sketch) $A D$ is the angle bisector of the angle at $A$ and $B H$ is the height from side $A C$. The obtuse angle between $B H$ and $A D$ is four times the size of angle $\angle D A B$. How big is the angle $\angle C A B$?
<image1>

Choices:
- $30^{\circ}$
- $45^{\circ}$
- $60^{\circ}$
- $75^{\circ}$
- $90^{\circ}$

metric geometry - angle

Question:
In triangle $A B C$ (see sketch) $A D$ is the angle bisector of the angle at $A$ and $B H$ is the height from side $A C$. The obtuse angle between $B H$ and $A D$ is four times the size of angle $\angle D A B$. How big is the angle $\angle C A B$?
<image1>
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\circ}$
(D) $75^{\circ}$
(E) $90^{\circ}$


Question:
How many blocks are missing in this igloo?
<image1>

Answer: 10

counting

Question:
How many blocks are missing in this igloo?
<image1>

Answer: 10


Question:
Amelia has a paper strip with five equal cells containing different drawings, according to the figure. She folds the strip in such a way that the cells overlap in five layers. Which of the sequences of layers, from top to bottom, is not possible to obtain?
<image1>
<image2>

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

topology

Question:
Amelia has a paper strip with five equal cells containing different drawings, according to the figure. She folds the strip in such a way that the cells overlap in five layers. Which of the sequences of layers, from top to bottom, is not possible to obtain?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Dennis takes off one of the squares of this shape <image1>
How many of these 5 shapes can he get?
<image2>

Answer: 3

combinatorial geometry

Question:
Dennis takes off one of the squares of this shape <image1>
How many of these 5 shapes can he get?
<image2>

Answer: 3


Question:
In triangle $ABC$ lines $CE$ and $AD$ are drawn so that

$\frac{CD}{DB}=\frac{3}{1}$ and $\frac{AE}{EB}=\frac{3}{2}$. Let $r=\frac{CP}{PE}$

where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:

<image1>

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

metric geometry - length

Question:
In triangle $ABC$ lines $CE$ and $AD$ are drawn so that

$\frac{CD}{DB}=\frac{3}{1}$ and $\frac{AE}{EB}=\frac{3}{2}$. Let $r=\frac{CP}{PE}$

where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:

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


Question: As shown in the diagram, $F G H I$ is a trapezium with side $G F$ parallel to $H I$. The lengths of $F G$ and $H I$ are 50 and 20 respectively. The point $J$ is on the side $F G$ such that the segment $I J$ divides the trapezium into two parts of equal area. What is the length of $F J$ ? <image1>

Answer: 35

Solution: Let $x$ be the length $F J$, and $h$ be the height of the trapezium. Then the area of triangle $F J I$ is $\frac{1}{2} x h$ and the area of trapezium FGHI is $\frac{1}{2} h(20+50)=35 h$. The triangle is half the area of the trapezium, so $\frac{1}{2} x h=\frac{1}{2} \times 35 h$, so $x=35$.

metric geometry - length

Question: As shown in the diagram, $F G H I$ is a trapezium with side $G F$ parallel to $H I$. The lengths of $F G$ and $H I$ are 50 and 20 respectively. The point $J$ is on the side $F G$ such that the segment $I J$ divides the trapezium into two parts of equal area. What is the length of $F J$ ? <image1>

Answer: 35


Question: Numbers are to be placed into the table shown, one number in each cell, in such a way that each row has the same total, and each column has the same total. Some of the numbers are already given. What number is $x$ ? <image1>

Answer: 4

Solution: Let the missing entry from the first row be $y$. Since the second and third columns have the same sum, the missing entry in the second column must be $y-3$. Then the first row adds to $8+y$, and the third row adds to $4+y+x$. Since these are the same sum, we must have $x=4$.

algebra

Question: Numbers are to be placed into the table shown, one number in each cell, in such a way that each row has the same total, and each column has the same total. Some of the numbers are already given. What number is $x$ ? <image1>

Answer: 4


Question: Square $A B C D$ has sides of length $3 \mathrm{~cm}$. The points $M$ and $N$ lie on $A D$ and $A B$ so that $C M$ and $C N$ split the square into three pieces of the same area. What is the length of $D M$ ? <image1>

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

Solution: The area of square $A B C D$ is $(3 \times 3) \mathrm{cm}^{2}=9 \mathrm{~cm}^{2}$. Hence the area of each piece is $\frac{1}{3} \times 9 \mathrm{~cm}^{2}=3 \mathrm{~cm}^{2}$. Since the area of a triangle is equal to half its base multiplied by its perpendicular height, we have $\frac{1}{2} \times D M \times D C=3 \mathrm{~cm}^{2}$. Therefore $\frac{1}{2} \times D M \times 3 \mathrm{~cm}=3 \mathrm{~cm}^{2}$ and hence the length of $D M$ is $2 \mathrm{~cm}$.

metric geometry - length

Question:
Square $A B C D$ has sides of length $3 \mathrm{~cm}$. The points $M$ and $N$ lie on $A D$ and $A B$ so that $C M$ and $C N$ split the square into three pieces of the same area. What is the length of $D M$ ? <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:
Let $a$ and $b$ be the lengths of the two shorter sides of the right-angled triangle shown in the diagram. The longest side, $D$, is the diameter of the large circle and $d$ is the diameter of the small circle, which touches all three sides of the triangle.
Which one of the following expressions is equal to $D+d$ ? <image1>

Choices:
- $(a+b)$
- $2(a+b)$
- $\frac{1}{2}(a+b)$
- $\sqrt{a b}$
- $\sqrt{a^{2}+b^{2}}$

Solution: The two tangents drawn from a point to a circle are equal in length, so we can mark the lengths of the tangents $r, s$ and $t$ on the diagram. Since the triangle is right-angled, and a tangent is perpendicular to the radius, through the point of contact, the small quadrilateral is a square, with all sides equal to $r$. Now $d=2 r$, so we have $D+d= (s+t)+2 r=(r+t)+(r+s)=a+b$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-039.jpg?height=365&width=623&top_left_y=1779&top_left_x=1179)

metric geometry - length

Question:
Let $a$ and $b$ be the lengths of the two shorter sides of the right-angled triangle shown in the diagram. The longest side, $D$, is the diameter of the large circle and $d$ is the diameter of the small circle, which touches all three sides of the triangle.
Which one of the following expressions is equal to $D+d$ ? <image1>
(A) $(a+b)$
(B) $2(a+b)$
(C) $\frac{1}{2}(a+b)$
(D) $\sqrt{a b}$
(E) $\sqrt{a^{2}+b^{2}}$


Question: Tom, John and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points, as shown. How many points did Lily score? <image1>

Answer: 40

Solution: Tom hit the inner ring three times, the middle ring once and the outer ring twice and scored 46 points. John hit the inner ring once, the middle ring three times and the outer ring twice and scored 34 points. Together they scored 80 points from hitting each ring four times. Lily hit each ring twice and so scored half of the total score of Tom and John. Therefore Lily scored 40 points.

algebra

Question: Tom, John and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points, as shown. How many points did Lily score? <image1>

Answer: 40


Question:
Maria made a block using white cubes and colored cubes in equal amounts. How many of the white cubes cannot be seen in the picture?
<image1>

Answer: 2

solid geometry

Question:
Maria made a block using white cubes and colored cubes in equal amounts. How many of the white cubes cannot be seen in the picture?
<image1>

Answer: 2


Question:
In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} = 3: 3: 1$, and similarly for lines $ BE$ and $ CF$. Then the area of triangle $ N_1N_2N_3$ is:
<image1>

Choices:
- $\frac{1}{10} \triangle ABC$
- $\frac{1}{9} \triangle ABC$
- $\frac{1}{7} \triangle ABC$
- $\frac{1}{6} \triangle ABC$
- $\text{none of these}$

metric geometry - area

Question:
In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} = 3: 3: 1$, and similarly for lines $ BE$ and $ CF$. Then the area of triangle $ N_1N_2N_3$ is:
<image1>
(A) $\frac{1}{10} \triangle ABC$
(B) $\frac{1}{9} \triangle ABC$
(C) $\frac{1}{7} \triangle ABC$
(D) $\frac{1}{6} \triangle ABC$
(E) $\text{none of these}$


Question: Each of $\triangle PQR$ and $\triangle STU$ has an area of $1.$ In $\triangle PQR,$ $U,$ $W,$ and $V$ are the midpoints of the sides. In $\triangle STU,$ $R,$ $V,$ and $W$ are the midpoints of the sides. What is the area of parallelogram $UVRW?$ <image1>

Answer: \frac{1}{2}

Solution:
Since $V$ is the midpoint of $PR,$ then $PV=VR.$  Since $UVRW$ is a parallelogram, then $VR=UW.$ Since $W$ is the midpoint of $US,$ then $UW=WS.$

Thus, $$PV=VR=UW=WS.$$ Similarly, $$QW=WR=UV=VT.$$ Also, $R$ is the midpoint of $TS$ and therefore, $TR=RS.$ Thus, $\triangle VTR$ is congruent to $\triangle WRS$, and so the two triangles have equal area.

Diagonal $VW$ in parallelogram $UVRW$ divides the area of the parallelogram in half. Therefore, $\triangle UVW$ and $\triangle RWV$ have equal areas.

In quadrilateral $VRSW,$ $VR=WS$ and $VR$ is parallel to $WS.$ Thus, $VRSW$ is a parallelogram and the area of $\triangle RWV$ is equal to the area of $\triangle WRS.$ Therefore, $\triangle VTR,$ $\triangle WRS,$ $\triangle RWV,$ and $\triangle UVW$ have equal areas, and so these four triangles divide $\triangle STU$ into quarters.

Parallelogram $UVRW$ is made from two of these four quarters of $\triangle STU,$ or one half of $\triangle STU.$ The area of parallelogram $UVRW$ is thus half of $1,$ or $\boxed{\frac{1}{2}}.$

metric geometry - area

Question: Each of $\triangle PQR$ and $\triangle STU$ has an area of $1.$ In $\triangle PQR,$ $U,$ $W,$ and $V$ are the midpoints of the sides. In $\triangle STU,$ $R,$ $V,$ and $W$ are the midpoints of the sides. What is the area of parallelogram $UVRW?$ <image1>

Answer: \frac{1}{2}


Question: Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$.\n<image1>

Answer: 59

Solution: Observe the pentagon as it fits in the following figure. The area we are looking for is one-third of the difference in areas of the equilateral triangles, which is $\frac{29^2 \sqrt{3} / 4-13^2 \sqrt{3} / 4}{3}=56 \sqrt{3}$. The answer is $56+3=59$.

metric geometry - area

Question: Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$.\n<image1>

Answer: 59


Question:
Rory uses four identical standard dice to build the solid shown in the diagram.
Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some of the faces of the solid are shown. What number is written on the face marked with question mark?
(On a standard die, the numbers on opposite faces add to 7.)
<image1>

Answer: 5

Solution: The information in the question tells us that the numbers on touching faces of the solid are the same and that numbers on opposite faces of a die add to 7 .  Since the number 4 is visible on the rear of the right-hand side of the solid, there is a 3 on the left-hand face of the  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f598bb42a47f3275d94g-3.jpg?height=225&width=354&top_left_y=1758&top_left_x=1411) rear right die and hence a 3 and a 4 on the right- and lefthand faces of the rear left die. Similarly, since the number 1 is visible on the left-hand side of the front of the solid, there is a 6 and a 1 on the front and back faces of the rear left die. Therefore the top and bottom faces of the rear left die have a 2 and a 5 written on them. Since the four dice are identical, comparison with the front right die of the solid tells us that a die with a 6 on its front face and a 3 on its right-hand face has a 2 on its lower face and hence a 5 on its upper face.

logic

Question:
Rory uses four identical standard dice to build the solid shown in the diagram.
Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some of the faces of the solid are shown. What number is written on the face marked with question mark?
(On a standard die, the numbers on opposite faces add to 7.)
<image1>

Answer: 5


Question:
The area of the rectangular region is

<image1>

Choices:
- $\text{.088 m}^2$
- $\text{.62 m}^2$
- $\text{.88 m}^2$
- $\text{1.24 m}^2$
- $\text{4.22 m}^2$

metric geometry - area

Question:
The area of the rectangular region is

<image1>
(A) $\text{.088 m}^2$
(B) $\text{.62 m}^2$
(C) $\text{.88 m}^2$
(D) $\text{1.24 m}^2$
(E) $\text{4.22 m}^2$


Question: The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure. Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$.\n<image1>

Answer: 100

Solution: \text { The total area is } 5 \pi-4 \times\left(\frac{\pi}{2}-1\right)=4+3 \pi \text {. }

metric geometry - area

Question: The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure. Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$.\n<image1>

Answer: 100


Question:
In the figure, it is given that angle $ C = 90^{\circ}, \overline{AD} = \overline{DB}, DE \perp AB, \overline{AB} = 20$, and $ \overline{AC} = 12$. The area of quadrilateral $ ADEC$ is:
<image1>

Choices:
- $75$
- $58\frac{1}{2}$
- $48$
- $37\frac{1}{2}$
- $\text{none of these}$

metric geometry - area

Question:
In the figure, it is given that angle $ C = 90^{\circ}, \overline{AD} = \overline{DB}, DE \perp AB, \overline{AB} = 20$, and $ \overline{AC} = 12$. The area of quadrilateral $ ADEC$ is:
<image1>
(A) $75$
(B) $58\frac{1}{2}$
(C) $48$
(D) $37\frac{1}{2}$
(E) $\text{none of these}$


Question:
Some of the small squares on each of the square transparencies have been coloured black. If you slide the three transparencies on top of each other, without lifting them from the table, a new pattern can be seen. What is the maximum number of black squares which could be seen in the new pattern?
<image1>

Answer: 8

descriptive geometry

Question:
Some of the small squares on each of the square transparencies have been coloured black. If you slide the three transparencies on top of each other, without lifting them from the table, a new pattern can be seen. What is the maximum number of black squares which could be seen in the new pattern?
<image1>

Answer: 8


Question:
How many points are there in the three unseen sides of dice?
<image1>

Answer: 11

arithmetic

Question:
How many points are there in the three unseen sides of dice?
<image1>

Answer: 11


Question:
Walter displayed all the integers from 0 to 109 according to some simple rule. Here is the beginning of his 5-column numeral chart. Which of the following elements could not be the a part of Walter's chart?
<image1>
<image2>

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

algebra

Question:
Walter displayed all the integers from 0 to 109 according to some simple rule. Here is the beginning of his 5-column numeral chart. Which of the following elements could not be the a part of Walter's chart?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
<image1>

Choices:
- $\frac{2}{7}$
- $\frac{1}{3}$
- $\frac{2}{3}$
- $\frac{3}{4}$
- $\frac{7}{9}$

analytic geometry

Question:
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
<image1>
(A) $\frac{2}{7}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$
(E) $\frac{7}{9}$


Question:
In the adjacent picture we see that $1+3+5+7=4 \times 4$. How big is $1+3+5+7+\ldots+17+19$ ?
<image1>

Choices:
- $10 \times 10$
- $11 \times 11$
- $12 \times 12$
- $13 \times 13$
- $14 \times 14$

algebra

Question:
In the adjacent picture we see that $1+3+5+7=4 \times 4$. How big is $1+3+5+7+\ldots+17+19$ ?
<image1>
(A) $10 \times 10$
(B) $11 \times 11$
(C) $12 \times 12$
(D) $13 \times 13$
(E) $14 \times 14$


Question:
Turning a card around on the top side, we see the photo of the kangaroo. Instead, if we turn the card around on the right side, what will appear?
<image1>
<image2>

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

transformation geometry

Question:
Turning a card around on the top side, we see the photo of the kangaroo. Instead, if we turn the card around on the right side, what will appear?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


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

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

descriptive geometry

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


Question:
What is the perimeter of the star (in centimetres) if you know that the star on the picture is formed by four equal circles with radius $5 \mathrm{~cm}$, one square and four equilateral triangles?
<image1>

Answer: 160

metric geometry - length

Question:
What is the perimeter of the star (in centimetres) if you know that the star on the picture is formed by four equal circles with radius $5 \mathrm{~cm}$, one square and four equilateral triangles?
<image1>

Answer: 160


Question:
In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$.  If $YZ = 12$, $\tan Z = 1.5$, and $\tan X = 3$, then what is the area of $WXYZ$?

<image1>

Answer: 162

Solution:
In order to find the area of trapezoid $WXYZ$, we must find the lengths of both bases and the distance (height) between those bases.

[asy]
pair WW,X,Y,Z;
Z = (0,0);
Y = (12,0);
WW = (12,18);
X= (18,18);
draw(WW--Y);
draw(rightanglemark(WW,Y,Z,30));
draw(rightanglemark(Y,WW,X,30));
draw(WW--X--Y--Z--WW);
label("$W$",WW,N);
label("$X$",X,N);
label("$Y$",Y,S);
label("$Z$",Z,S);
label("$12$",Y/2,S);
[/asy]

We add $\overline{WY}$ to our diagram and note that because $\overline{WX}\parallel\overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$, we have $\overline{WY}\perp\overline{WX}$.  Therefore, triangles $WYX$ and $WYZ$ are right triangles.

From right triangle $WYZ$, we have $\tan Z = \frac{WY}{ZY}$, so $WY = ZY\cdot \tan Z = 12\cdot 1.5 = 18$.  From right triangle $WXY$, we have $\tan X = \frac{WY}{WX}$, so \[WX = \frac{WY}{\tan X} = \frac{18}{3} = 6.\]Therefore, the area of $WXYZ$ is \[\frac{WX+YZ}{2}\cdot WY = 9\cdot 18 = \boxed{162}.\]

metric geometry - area

Question:
In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$.  If $YZ = 12$, $\tan Z = 1.5$, and $\tan X = 3$, then what is the area of $WXYZ$?

<image1>

Answer: 162


Question:
Jenny looks at her weather app that shows the predicted weather and maximum temperatures for the next five days. Which of the following represents the corresponding graph of maximum temperatures?
<image1>
<image2>

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

statistics

Question:
Jenny looks at her weather app that shows the predicted weather and maximum temperatures for the next five days. Which of the following represents the corresponding graph of maximum temperatures?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: Consider the figure below, not drawn to scale.\nIn this figure, assume that$AB \perp BE$ and $AD  \perp DE$. Also, let $AB = \sqrt{6}$ and $\angle BED =\frac{\pi}{6}$ . Find $AC$.\n<image1>

Answer: $2\sqrt{2}$

metric geometry - length

Question: Consider the figure below, not drawn to scale.\nIn this figure, assume that$AB \perp BE$ and $AD  \perp DE$. Also, let $AB = \sqrt{6}$ and $\angle BED =\frac{\pi}{6}$ . Find $AC$.\n<image1>

Answer: $2\sqrt{2}$


Question: What is the sum of the 10 angles marked on the diagram on the right? <image1>

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

Solution: The sum of the fifteen angles in the five triangles is $5 \times 180^{\circ}=900^{\circ}$. The sum of the unmarked central angles in the five triangles is $180^{\circ}$, since each can be paired with the angle between the two triangles opposite. Thus the sum of the marked angles is $900^{\circ}-180^{\circ}=720^{\circ}$.

metric geometry - angle

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


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

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

combinatorial geometry

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


Question:
Each of the nets of a cube has a line drawn on. For which net does the line form a closed loop when the net is folded up to make a cube?
<image1>

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

descriptive geometry

Question:
Each of the nets of a cube has a line drawn on. For which net does the line form a closed loop when the net is folded up to make a cube?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The sides of the rectangle $A B C D$ are parallel to the co-ordinate axes. The rectangle is positioned below the $x$-axis and to the right of the $y$-axis, as shown in the picture. The co-ordinates of the points $A, B, C, D$ are whole numbers. For each of the points we calculate the value of ( $y$ co-ordinate) $\div$(x co-ordinate). Which of the points will give the smallest value?
<image1>

Choices:
- A
- B
- C
- D
- It depends on the rectangle.

analytic geometry

Question:
The sides of the rectangle $A B C D$ are parallel to the co-ordinate axes. The rectangle is positioned below the $x$-axis and to the right of the $y$-axis, as shown in the picture. The co-ordinates of the points $A, B, C, D$ are whole numbers. For each of the points we calculate the value of ( $y$ co-ordinate) $\div$(x co-ordinate). Which of the points will give the smallest value?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) It depends on the rectangle.


Question:
How many more bricks does the right hand pyramid have than the left hand pyramid?
<image1>

Answer: 5

counting

Question:
How many more bricks does the right hand pyramid have than the left hand pyramid?
<image1>

Answer: 5


Question: Each cell of a $3 \times 3$ square has a number written in it. Unfortunately the numbers are not visible because they are covered in ink. However, the sum of the numbers in each row and the sum of the numbers in two of the columns are all known, as shown by the arrows on the diagram. What is the sum of the numbers in the third column? <image1>

Answer: 43

Solution: The sum of the row totals is the sum of all the nine numbers in the $3 \times 3$ square. Likewise, the sum of the column totals is the sum of these nine numbers. Therefore $24+26+40=27+20+x$, where $x$ is the sum of the numbers in the third column. Therefore $90=47+x$ and hence $x=43$. Therefore the sum of the numbers in the third column is 43 .

algebra

Question: Each cell of a $3 \times 3$ square has a number written in it. Unfortunately the numbers are not visible because they are covered in ink. However, the sum of the numbers in each row and the sum of the numbers in two of the columns are all known, as shown by the arrows on the diagram. What is the sum of the numbers in the third column? <image1>

Answer: 43


Question:
What is the area of the shaded region of the given $8 \times 5$ rectangle?

<image1>

Choices:
- $4\frac{3}{5}$
- $5$
- $5\frac{1}{4}$
- $6\frac{1}{2}$
- $8$

metric geometry - area

Question:
What is the area of the shaded region of the given $8 \times 5$ rectangle?

<image1>
(A) $4\frac{3}{5}$
(B) $5$
(C) $5\frac{1}{4}$
(D) $6\frac{1}{2}$
(E) $8$


Question:
<image1>

A bar graph shows the number of hamburgers sold by a fast food chain each season.  However, the bar indicating the number sold during the winter is covered by a smudge.  If exactly $ 25 \%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?

Answer: 2.5

statistics

Question:
<image1>

A bar graph shows the number of hamburgers sold by a fast food chain each season.  However, the bar indicating the number sold during the winter is covered by a smudge.  If exactly $ 25 \%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?

Answer: 2.5


Question:
The regular pentagon $P Q R S T$ in the diagram has been reflected in the line $P Q$ so that vertex $T$ is reflected to point $U$, as shown. Then the new pentagon is reflected in $P U$, so that vertex $Q$ is reflected to point $V$, as shown. This process is repeated, on each occasion reflecting in the line determined by the new edge through $P$.
What is the least number of such reflections that are needed to return pentagon $P Q R S T$ to its original position?
<image1>

Answer: 10

Solution: Note that $P$ does not move and that two successive reflections of the pentagon are equivalent to clockwise rotation about $P$ of $2 \times 108^{\circ}=216^{\circ}$. To return to the original position, the pentagon needs to be rotated through an integer multiple of $360^{\circ}$. The lowest common multiple of 216 and 360 is 1080 , corresponding to ten reflections.

transformation geometry

Question:
The regular pentagon $P Q R S T$ in the diagram has been reflected in the line $P Q$ so that vertex $T$ is reflected to point $U$, as shown. Then the new pentagon is reflected in $P U$, so that vertex $Q$ is reflected to point $V$, as shown. This process is repeated, on each occasion reflecting in the line determined by the new edge through $P$.
What is the least number of such reflections that are needed to return pentagon $P Q R S T$ to its original position?
<image1>

Answer: 10


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

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

metric geometry - area

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


Question: In the diagram, $A B C D$ is a parallelogram. If $A A_{1}=4 \mathrm{~cm}, D D_{1}=5 \mathrm{~cm}$ and $C C_{1}=7 \mathrm{~cm}$, what is the length of $B B_{1}$ ? <image1>

Choices:
- $9 \mathrm{~cm}$
- $11 \mathrm{~cm}$
- $12 \mathrm{~cm}$
- $16 \mathrm{~cm}$
- $21 \mathrm{~cm}$

Solution: Construct a rectangle $P Q R S$ around the parallelogram $A B C D$ so that $P S$ is parallel to $A_{1} C_{1}$. Because triangles $A B Q$ and $C D S$ are congruent, $A Q=C S=$ $7+5 \mathrm{~cm}=12 \mathrm{~cm}$ and so $B B_{1}=A Q+A A_{1}=12+4 \mathrm{~cm}$ $=16 \mathrm{~cm}$.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-025.jpg?height=585&width=748&top_left_y=701&top_left_x=1042)

metric geometry - length

Question:
In the diagram, $A B C D$ is a parallelogram. If $A A_{1}=4 \mathrm{~cm}, D D_{1}=5 \mathrm{~cm}$ and $C C_{1}=7 \mathrm{~cm}$, what is the length of $B B_{1}$ ? <image1>
(A) $9 \mathrm{~cm}$
(B) $11 \mathrm{~cm}$
(C) $12 \mathrm{~cm}$
(D) $16 \mathrm{~cm}$
(E) $21 \mathrm{~cm}$


Question:
If a laser beam hits a mirror it changes its direction (see left diagram). Each mirror has mirrored sides on both sides. At which letter does the laser beam end?
<image1>

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

transformation geometry

Question:
If a laser beam hits a mirror it changes its direction (see left diagram). Each mirror has mirrored sides on both sides. At which letter does the laser beam end?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Which of the five squares has the biggest proportion of black area?
<image1>

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

combinatorial geometry

Question:
Which of the five squares has the biggest proportion of black area?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
In the diagram on the right we want to colour the fields with the colours A, B, C and D so that adjacent fields are always in different colours. (Even fields that share only one corner, count as adjacent.) Some fields have already been coloured in. In which colour can the grey field be coloured in?
<image1>

Choices:
- either A or B
- only C
- only D
- either C or D
- A, B, C or D

graph theory

Question:
In the diagram on the right we want to colour the fields with the colours A, B, C and D so that adjacent fields are always in different colours. (Even fields that share only one corner, count as adjacent.) Some fields have already been coloured in. In which colour can the grey field be coloured in?
<image1>
(A) either A or B
(B) only C
(C) only D
(D) either C or D
(E) A, B, C or D


Question:
Which of the following polygons has the largest area?

<image1>

Choices:
- $\text{A}$
- $\text{B}$
- $\text{C}$
- $\text{D}$
- $\text{E}$

combinatorial geometry

Question:
Which of the following polygons has the largest area?

<image1>
(A) $\text{A}$
(B) $\text{B}$
(C) $\text{C}$
(D) $\text{D}$
(E) $\text{E}$


Question:
Northside's Drum and Bugle Corps raised money for a trip.  The drummers and bugle players kept separate sales records.  According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?

<image1>

Choices:
- $\text{Jan}$
- $\text{Feb}$
- $\text{Mar}$
- $\text{Apr}$
- $\text{May}$

statistics

Question:
Northside's Drum and Bugle Corps raised money for a trip.  The drummers and bugle players kept separate sales records.  According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?

<image1>
(A) $\text{Jan}$
(B) $\text{Feb}$
(C) $\text{Mar}$
(D) $\text{Apr}$
(E) $\text{May}$


Question: A dark disc with two holes is placed on the dial of a watch as shown in the diagram. The dark disc is now rotated so that the number 10 can be seen through one of the two holes. Which of the numbers could one see through the other hole now? <image1>

Choices:
- 2 and 6
- 3 and 7
- 3 and 6
- 1 and 9
- 2 and 7

transformation geometry

Question:
A dark disc with two holes is placed on the dial of a watch as shown in the diagram. The dark disc is now rotated so that the number 10 can be seen through one of the two holes. Which of the numbers could one see through the other hole now? <image1>
(A) 2 and 6
(B) 3 and 7
(C) 3 and 6
(D) 1 and 9
(E) 2 and 7


Question:
Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.
<image1>
\section*{ACROSS}
1. Four less than a factor of 105.
3. One more than a palindrome.
5. The square-root of the answer to this Kangaroo question.
\section*{DOWN}
1. Two less than a square.
2. Four hundred less than a cube.
4. Six less than the sum of the answers to two of the other clues.
What is the square of the answer to 5 ACROSS?

Answer: 841

Solution: The two-digit factors of 105 are 15,21 and 35 , so 1 ACROSS is one of 11,17 or 31. However, as 2 DOWN is 400 less than a cube it cannot start with a 7, so 1 ACROSS is either 11 or 31 . If 1 ACROSS is 11 then 1 DOWN is 14, 2 DOWN is 112 and 3 ACROSS is 415 . This means 4 DOWN is at least 50 which is too large to satisfy its clue, since 5 ACROSS is at most 29. So 1 ACROSS is 31. This means 1 DOWN is 34, 2 DOWN is 112 and 3 ACROSS is 415. Checking each of the digits 1 to 9 in turn for the units digit of 4 DOWN yields 9 as the only possibility, with $59=31+34-6$. Therefore, 5 ACROSS is 29 and its square is 841.

logic

Question:
Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.
<image1>
\section*{ACROSS}
1. Four less than a factor of 105.
3. One more than a palindrome.
5. The square-root of the answer to this Kangaroo question.
\section*{DOWN}
1. Two less than a square.
2. Four hundred less than a cube.
4. Six less than the sum of the answers to two of the other clues.
What is the square of the answer to 5 ACROSS?

Answer: 841


Question:
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11$. With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
<image1>

Answer: 640

combinatorics

Question:
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11$. With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
<image1>

Answer: 640


Question:
A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

<image1>

Choices:
- $4 \sqrt{3}$
- $\frac{120}{17}$
- $10$
- $\frac{17\sqrt{2}}{2}$
- $\frac{17\sqrt{3}}{2}$

metric geometry - length

Question:
A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

<image1>
(A) $4 \sqrt{3}$
(B) $\frac{120}{17}$
(C) $10$
(D) $\frac{17\sqrt{2}}{2}$
(E) $\frac{17\sqrt{3}}{2}$


Question: Three congruent isosceles triangles $DAO,$ $AOB,$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12.$ These triangles are arranged to form trapezoid $ABCD,$ as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB.$ <image1> What is the length of $OP?$

Answer: 8

Solution: Since $\triangle AOB$ is isosceles with $AO=OB$ and $OP$ is perpendicular to $AB,$ then $P$ is the midpoint of $AB,$ so $$AP=PB=\frac{1}{2}AB=\frac{1}{2}(12)=6.$$ By the Pythagorean Theorem, $OP = \sqrt{AO^2 - AP^2},$ so we have $$OP = \sqrt{10^2-6^2}=\sqrt{64}=\boxed{8}.$$

metric geometry - length

Question: Three congruent isosceles triangles $DAO,$ $AOB,$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12.$ These triangles are arranged to form trapezoid $ABCD,$ as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB.$ <image1> What is the length of $OP?$

Answer: 8


Question:
The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area.  How many inches long is the side of the center square?

<image1>

Answer: 50

combinatorial geometry

Question:
The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area.  How many inches long is the side of the center square?

<image1>

Answer: 50


Question:
The numbers from 1 to 6 are placed in the circles at the intersections of 3 rings. The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?
<image1>

Answer: 1

logic

Question:
The numbers from 1 to 6 are placed in the circles at the intersections of 3 rings. The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?
<image1>

Answer: 1


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

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

statistics

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


Question:
The picture on the right shows a tile pattern. The side length of the bigger tiles is a and of the smaller ones b. The dotted lines (horizontal and tilted) include an angle of $30^{\circ}$. How big is the ratio a:b?
<image1>

Choices:
- $(2 \cdot \sqrt{3}): 1$
- $(2+\sqrt{3}): 1$
- $(3+\sqrt{2}): 1$
- $(3 \cdot \sqrt{2}): 1$
- 2:1

metric geometry - angle

Question:
The picture on the right shows a tile pattern. The side length of the bigger tiles is a and of the smaller ones b. The dotted lines (horizontal and tilted) include an angle of $30^{\circ}$. How big is the ratio a:b?
<image1>
(A) $(2 \cdot \sqrt{3}): 1$
(B) $(2+\sqrt{3}): 1$
(C) $(3+\sqrt{2}): 1$
(D) $(3 \cdot \sqrt{2}): 1$
(E) 2:1


Question: In right triangle $ABC$, $M$ and $N$ are midpoints of legs $\overline{AB}$ and $\overline{BC}$, respectively. Leg $\overline{AB}$ is 6 units long, and leg $\overline{BC}$ is 8 units long. How many square units are in the area of $\triangle APC$? <image1>

Answer: 8

Solution:
[asy]
draw((0,0)--(8,0)--(0,6)--cycle);
draw((0,0)--(4,3));
draw((4,0)--(0,6));
draw((0,3)--(8,0));
label("$A$",(0,6),NW); label("$B$",(0,0),SW); label("$C$",(8,0),SE); label("$M$",(0,3),W); label("$N$",(4,0),S); label("$P$",(8/3,2),N);
[/asy]

Drawing the three medians of a triangle divides the triangle into six triangles with equal area.  Triangle $APC$ consists of two of these triangles, so $[APC] = [ABC]/3 = (6\cdot 8/2)/3 = \boxed{8}$.

metric geometry - area

Question: In right triangle $ABC$, $M$ and $N$ are midpoints of legs $\overline{AB}$ and $\overline{BC}$, respectively. Leg $\overline{AB}$ is 6 units long, and leg $\overline{BC}$ is 8 units long. How many square units are in the area of $\triangle APC$? <image1>

Answer: 8


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

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

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

analytic geometry

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


Question:
Denis ties his dog, using an 11-meter rope, one meter away from a corner of about 7 meters by 5 meters, as illustrated. Denis places 5 bones near the fence, as shown in the picture. How many bones can the dog catch?
<image1>

Answer: 5

metric geometry - length

Question:
Denis ties his dog, using an 11-meter rope, one meter away from a corner of about 7 meters by 5 meters, as illustrated. Denis places 5 bones near the fence, as shown in the picture. How many bones can the dog catch?
<image1>

Answer: 5


Question:
Each shelf holds a total of 64 deciliters of apple juice. The bottles have three different sizes: large, medium and small. How many deciliters of apple juice does a medium bottle contain?
<image1>

Answer: 10

algebra

Question:
Each shelf holds a total of 64 deciliters of apple juice. The bottles have three different sizes: large, medium and small. How many deciliters of apple juice does a medium bottle contain?
<image1>

Answer: 10


Question: A large square is divided into smaller squares. In one of the smaller squares a diagonal is also drawn, as shown. What fraction of the large square is shaded? <image1>

Choices:
- $\frac{4}{5}$
- $\frac{3}{8}$
- $\frac{4}{9}$
- $\frac{1}{3}$
- $\frac{1}{2}$

Solution: The shaded square in the lower right corner of the large square is $\frac{1}{4}$ of the large square. The shaded triangle is half of $\frac{1}{4}$ of the large square. Hence it is $\frac{1}{8}$ of the large square. The two small shaded squares in the upper left corner together are half of $\frac{1}{4}$, or $\frac{1}{8}$, of the large square. Therefore the fraction of the large square that is shaded is $\frac{1}{4}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}$.

metric geometry - area

Question:
A large square is divided into smaller squares. In one of the smaller squares a diagonal is also drawn, as shown. What fraction of the large square is shaded? <image1>
(A) $\frac{4}{5}$
(B) $\frac{3}{8}$
(C) $\frac{4}{9}$
(D) $\frac{1}{3}$
(E) $\frac{1}{2}$


Question:
Maria colours exactly 5 cells of this grid <image1> in grey. Then she has her 5 friends guess which cells she has coloured in and their answers are the five patterns $A, B, C, D$ and $E$. Maria looks at the patterns and says: "One of you is right. The others have each guessed exactly four cells correctly." Which pattern did Maria paint?
<image2>

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

logic

Question:
Maria colours exactly 5 cells of this grid <image1> in grey. Then she has her 5 friends guess which cells she has coloured in and their answers are the five patterns $A, B, C, D$ and $E$. Maria looks at the patterns and says: "One of you is right. The others have each guessed exactly four cells correctly." Which pattern did Maria paint?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
<image1>

In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals

Choices:
- $\frac{1}{4}\cos 10^\circ$
- $\frac{\sqrt{3}}{8}$
- $\frac{1}{4}\cos 40^\circ$
- $\frac{1}{4}\cos 50^\circ$
- $\frac{1}{8}$

metric geometry - area

Question:
<image1>

In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals
(A) $\frac{1}{4}\cos 10^\circ$
(B) $\frac{\sqrt{3}}{8}$
(C) $\frac{1}{4}\cos 40^\circ$
(D) $\frac{1}{4}\cos 50^\circ$
(E) $\frac{1}{8}$


Question:
Mara built the square by using 4 of the following 5 shapes. Which shape was not used?
<image1>
<image2>

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

combinatorial geometry

Question:
Mara built the square by using 4 of the following 5 shapes. Which shape was not used?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question: $ABCD$ is a square with $AB = 8$cm. Arcs $BC$ and $CD$ are semicircles. Express the area of the shaded region, in square centimeters, and in terms of $\pi$. (As always, do not include units in your submitted answer.) <image1>

Answer: 8\pi-16

Solution:
[asy]
real y = .866025404;
draw((-1,-1)--(1, -1) -- (1, 1) -- (-1, 1)--cycle );
draw( Arc( (1,0), 1, 90, 270));
draw( Arc( (0,1), 1, 180, 360));
fill((0,0)..(.5, y).. (1,1) --cycle, blue); fill((0,0)..(y, .5).. (1,1) --cycle, gray(0.7));
label ("$A$", (-1,-1) , SW); label ("$B$", (1,-1) , SE); label ("$C$", (1,1) , NE); label ("$D$", (-1,1) , NW);
draw((0,0)--(1,0),blue+linewidth(.8));
label ("$X$", (0,0) , SW); label ("$Y$", (1,0) , SE);
[/asy]

Let the intersection of the arcs be $X$.  Draw line $XC$, which divides the shaded region into two equal shaded regions, one blue and one grey.  We will calculate the area of the blue region and multiply it by 2 to yield the total shaded area.

To calculate the area of the blue region, notice that it is equal to the area of the quarter circle bounded by arc $CX$, centered at $Y$, minus the area of triangle $\triangle CXY$.  The quarter circle has radius $8/2=4$ and area \[\frac{1}{4} \cdot \pi (4)^2 = 4\pi.\]The triangle has area \[\frac{1}{2} (4)(4) = 8.\]Hence the blue region has area $4\pi - 8$.  The entire shaded region has area twice this, which is $2(4\pi-8) = \boxed{8\pi - 16}$.

metric geometry - area

Question: $ABCD$ is a square with $AB = 8$cm. Arcs $BC$ and $CD$ are semicircles. Express the area of the shaded region, in square centimeters, and in terms of $\pi$. (As always, do not include units in your submitted answer.) <image1>

Answer: 8\pi-16


Question:
What is the piece that fits completely to the given one to form a rectangle?
<image1>
<image2>

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

combinatorial geometry

Question:
What is the piece that fits completely to the given one to form a rectangle?
<image1>
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
The sides of the rectangle $A B C D$ are parallel to the co-ordinate axes. The rectangle is positioned below the $x$-axis and to the right of the $y$-axis, as shown in the picture. The co-ordinates of the points $A, B, C, D$ are whole numbers. For each of the points we calculate the value of ( $y$ co-ordinate) $\div$(x co-ordinate). Which of the points will give the smallest value?
<image1>

Choices:
- A
- B
- C
- D
- It depends on the rectangle.

analytic geometry

Question:
The sides of the rectangle $A B C D$ are parallel to the co-ordinate axes. The rectangle is positioned below the $x$-axis and to the right of the $y$-axis, as shown in the picture. The co-ordinates of the points $A, B, C, D$ are whole numbers. For each of the points we calculate the value of ( $y$ co-ordinate) $\div$(x co-ordinate). Which of the points will give the smallest value?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) It depends on the rectangle.


Question:
Lisa has mounted 7 postcards on her fridge door using 8 strong magnets (black dots). What is the maximum amount of magnets she can remove without any postcards falling on the floor?
<image1>

Answer: 4

descriptive geometry

Question:
Lisa has mounted 7 postcards on her fridge door using 8 strong magnets (black dots). What is the maximum amount of magnets she can remove without any postcards falling on the floor?
<image1>

Answer: 4


Question:
The "tower" in the diagram on the left is made up of a sqaure, a rectangle and an equlateral triangle. Each of those three shapes has the same perimeter. The side length of the square is $9 \mathrm{~cm}$. How long is the side of the rectangle indicated?
<image1>

Choices:
- $4 \mathrm{~cm}$
- $5 \mathrm{~cm}$
- $6 \mathrm{~cm}$
- $7 \mathrm{~cm}$
- $8 \mathrm{~cm}$

metric geometry - length

Question:
The "tower" in the diagram on the left is made up of a sqaure, a rectangle and an equlateral triangle. Each of those three shapes has the same perimeter. The side length of the square is $9 \mathrm{~cm}$. How long is the side of the rectangle indicated?
<image1>
(A) $4 \mathrm{~cm}$
(B) $5 \mathrm{~cm}$
(C) $6 \mathrm{~cm}$
(D) $7 \mathrm{~cm}$
(E) $8 \mathrm{~cm}$


Question:
Different figures represent different digits. Find the digit corresponding to the square.
<image1>

Answer: 6

algebra

Question:
Different figures represent different digits. Find the digit corresponding to the square.
<image1>

Answer: 6


Question: The point $A(3,3)$ is reflected across the $x$-axis to $A^{'}$. Then $A^{'}$ is translated two units to the left to $A^{''}$. The coordinates of $A^{''}$ are $(x,y)$. What is the value of $x+y$? <image1>

Answer: -2

Solution:
After reflection, the point $A(3,3)$ becomes $A'(3,-3)$. After translation, the point $A'(3,-3)$ becomes $A''(1,-3)$. Thus, $x+y$ is equal to $1+(-3)=\boxed{-2}$. A picture of the transformations is below.

[asy]
Label f;

f.p=fontsize(6);

xaxis(0,4,Ticks(f, 1.0));

yaxis(-4,4,Ticks(f, 1.0));

dot((3,3));
dot((3,-3));
dot((1,-3));
label("$A$", (3,3), W);
label("$A'$", (3,-3), W);
label("$A''$", (1,-3), W);
[/asy]

analytic geometry

Question: The point $A(3,3)$ is reflected across the $x$-axis to $A^{'}$. Then $A^{'}$ is translated two units to the left to $A^{''}$. The coordinates of $A^{''}$ are $(x,y)$. What is the value of $x+y$? <image1>

Answer: -2


Question: Carl built the shape shown using seven unit cubes. How many such cubes does he have to add to make a cube with edges of length 3 ? <image1>

Answer: 20

Solution: A cube with edges of length 3 is made up of $3^{3}=27$ unit cubes. Carl has already used seven cubes to build the initial shape so the number of cubes he needs to add is $27-7=20$.

solid geometry

Question: Carl built the shape shown using seven unit cubes. How many such cubes does he have to add to make a cube with edges of length 3 ? <image1>

Answer: 20


Question:
Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is

<image1>

Answer: 7

metric geometry - area

Question:
Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is

<image1>

Answer: 7


Question:
In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. Write the perimeter of $\triangle BDP$ in simplest form as:  $w + \frac{x \cdot \sqrt{y}}{z}$, where $w, x, y, z$ are nonnegative integers.  What is $w + x + y + z$?

<image1>

Answer: 12

Solution:
Both triangles $APD$ and $CBD$ are 30-60-90 triangles.  Thus $DP=\frac{2\sqrt{3}}{3}$ and $DB=2$. Since $\angle
BDP=\angle PBD$, it follows that $PB=PD=\frac{2\sqrt{3}}{3}$. Hence the perimeter of $\triangle BDP$ is $\frac{2\sqrt{3}}{3}+\frac{2\sqrt{3}}{3}+2={2+\frac{4\sqrt{3}}{3}}$.  Converting this to our contrived notation gives $w + x + y + z = \boxed{12}$.

metric geometry - length

Question:
In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. Write the perimeter of $\triangle BDP$ in simplest form as:  $w + \frac{x \cdot \sqrt{y}}{z}$, where $w, x, y, z$ are nonnegative integers.  What is $w + x + y + z$?

<image1>

Answer: 12


Question:
In the figure $\alpha=7^{\circ}$. All lines $\mathrm{OA}_{1}, \mathrm{~A}_{1} \mathrm{~A}_{2}, \mathrm{~A}_{2} \mathrm{~A}_{3}, \ldots$ are equally long. What is the maximum number of lines that can be drawn in this way if no two lines are allowed to intersect each other?
<image1>

Answer: 13

metric geometry - length

Question:
In the figure $\alpha=7^{\circ}$. All lines $\mathrm{OA}_{1}, \mathrm{~A}_{1} \mathrm{~A}_{2}, \mathrm{~A}_{2} \mathrm{~A}_{3}, \ldots$ are equally long. What is the maximum number of lines that can be drawn in this way if no two lines are allowed to intersect each other?
<image1>

Answer: 13


Question:
Two $1 \mathrm{~cm}$ long segments are marked on opposite sides of a square with side length 8 $\mathrm{cm}$. The end points of the segments are connected with each other as shown in the diagram. How big is the area of the grey part?
<image1>

Choices:
- $2 \mathrm{~cm}^{2}$
- $4 \mathrm{~cm}^{2}$
- $6.4 \mathrm{~cm}^{2}$
- $8 \mathrm{~cm}^{2}$
- $10 \mathrm{~cm}^{2}$

metric geometry - area

Question:
Two $1 \mathrm{~cm}$ long segments are marked on opposite sides of a square with side length 8 $\mathrm{cm}$. The end points of the segments are connected with each other as shown in the diagram. How big is the area of the grey part?
<image1>
(A) $2 \mathrm{~cm}^{2}$
(B) $4 \mathrm{~cm}^{2}$
(C) $6.4 \mathrm{~cm}^{2}$
(D) $8 \mathrm{~cm}^{2}$
(E) $10 \mathrm{~cm}^{2}$


Question:
A building is made up of cubes of the same size. The three pictures show it from above (von oben), from the front (von vorne) and from the right (von rechts). What is the maximum number of cubes used to make this building?
<image1>

Answer: 19

descriptive geometry

Question:
A building is made up of cubes of the same size. The three pictures show it from above (von oben), from the front (von vorne) and from the right (von rechts). What is the maximum number of cubes used to make this building?
<image1>

Answer: 19


Question:
A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?
<image1>

Answer: 49

metric geometry - area

Question:
A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?
<image1>

Answer: 49


Question:
A paper cone is to be made from a three-quarter circle having radius 4 inches (shaded). What is the length of the arc on the discarded quarter-circle (dotted portion)? Express your answer in terms of $\pi$.

<image1>

Answer: 2\pi

Solution: The total circumference of the original circle was $2\cdot 4 \cdot \pi = 8\pi$.  The quarter-circle has length that is $\frac{1}{4}$ of this, or $\frac{1}{4}\cdot 8\pi = \boxed{2\pi}$ inches.

metric geometry - angle

Question:
A paper cone is to be made from a three-quarter circle having radius 4 inches (shaded). What is the length of the arc on the discarded quarter-circle (dotted portion)? Express your answer in terms of $\pi$.

<image1>

Answer: 2\pi


Question: Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the Primle. For each guess, a digit is highlighted blue if it is in the Primle, but not in the correct place. A digit is highlighted orange if it is in the Primle and is in the correct place. Finally, a digit is left unhighlighted if it is not in the Primle. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the Primle?\n<image1>

Answer: 79

Solution: From the second row, we know that the first digit of the prime has the be $7$. So, the possible primes are $71, 73,$ and $79$. But, we know that neither $1$ nor $3$ can be in the prime number from the information in the first row. Therefore, the Primle is $\boxed{79}$.

logic

Question: Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the Primle. For each guess, a digit is highlighted blue if it is in the Primle, but not in the correct place. A digit is highlighted orange if it is in the Primle and is in the correct place. Finally, a digit is left unhighlighted if it is not in the Primle. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the Primle?\n<image1>

Answer: 79


Question:
A parallelogram is divided in two parts $P_{1}$ and $P_{2}$, as shown in the picture. Which sentence is always true?
<image1>

Choices:
- $P_{2}$ has a longer perimeter than $P_{1}$
- $P_{2}$ has a smaller perimeter than $P_{1}$
- $P_{2}$ has a smaller area than $P_{1}$
- $P_{1}$ and $P_{2}$ have the same perimeter
- $P_{1}$ and $P_{2}$ have the same area

transformation geometry

Question:
A parallelogram is divided in two parts $P_{1}$ and $P_{2}$, as shown in the picture. Which sentence is always true?
<image1>
(A) $P_{2}$ has a longer perimeter than $P_{1}$
(B) $P_{2}$ has a smaller perimeter than $P_{1}$
(C) $P_{2}$ has a smaller area than $P_{1}$
(D) $P_{1}$ and $P_{2}$ have the same perimeter
(E) $P_{1}$ and $P_{2}$ have the same area


Question:
An underground line has the six stations A, B, C, D, E and F. The train stops at every station. After reaching the end of the line $A$ or $F$ the train continues in the opposite direction. The train conductor starts his journey in station B. His first stop is in station C. In which station will be his $46^{\text {th}}$ stop?
<image1>

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

algebra

Question:
An underground line has the six stations A, B, C, D, E and F. The train stops at every station. After reaching the end of the line $A$ or $F$ the train continues in the opposite direction. The train conductor starts his journey in station B. His first stop is in station C. In which station will be his $46^{\text {th}}$ stop?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
An equilateral triangle and a regular hexagon are inscribed in a circle, the latter beeing inscribed in an equilateral triangle (see the picture). $S$ is the area of the big triangle, $s$ the area of the little one and $Q$ is the area of the hexagon. What is true?
<image1>

Choices:
- $Q=\sqrt{S \cdot s}$
- $Q=\frac{S+s}{2}$
- $S=s+Q$
- $Q=\sqrt{S^{2}+s^{2}}$
- $S=Q+3 s$

metric geometry - area

Question:
An equilateral triangle and a regular hexagon are inscribed in a circle, the latter beeing inscribed in an equilateral triangle (see the picture). $S$ is the area of the big triangle, $s$ the area of the little one and $Q$ is the area of the hexagon. What is true?
<image1>
(A) $Q=\sqrt{S \cdot s}$
(B) $Q=\frac{S+s}{2}$
(C) $S=s+Q$
(D) $Q=\sqrt{S^{2}+s^{2}}$
(E) $S=Q+3 s$


Question:
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
<image1>

Answer: 8

combinatorial geometry

Question:
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
<image1>

Answer: 8


Question: There are five big trees and three paths in a park. It has been decided to plant a sixth tree so that there are the same number of trees on either side of each path. In which region of the park should the sixth tree be planted? <image1>

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

Solution: The path running from the top of the park to the bottom has two trees to the left of it and three trees to the right of it on the diagram. Hence the sixth tree should be planted to the left of this path. The path running from the top left of the park to the bottom right has two trees above it and three trees below it on the diagram. Hence the sixth tree should be planted above this path. When we combine these observations, we can see that the sixth tree should be planted in the region labelled B. Note: this would also mean that there were the same number of trees on either side of the third path.

logic

Question:
There are five big trees and three paths in a park. It has been decided to plant a sixth tree so that there are the same number of trees on either side of each path. In which region of the park should the sixth tree be planted? <image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
Elvis has 6 triangles with this pattern
<image1>
Which picture can he make with them?
<image2>

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

combinatorial geometry

Question:
Elvis has 6 triangles with this pattern
<image1>
Which picture can he make with them?
<image2>
(A) A
(B) B
(C) C
(D) D
(E) E


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:
A barber wants to write the word SHAVE on a board so that a customer who sees the word in the mirror can read the word normally. How does he have to write the word on the board?
<image1>

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

transformation geometry

Question:
A barber wants to write the word SHAVE on a board so that a customer who sees the word in the mirror can read the word normally. How does he have to write the word on the board?
<image1>
(A) A
(B) B
(C) C
(D) D
(E) E


Question:
If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is

<image1>

Choices:
- $\frac{1}{3}$
- $\frac{2}{5}$
- $\frac{7}{15}$
- $\frac{8}{15}$
- $\frac{3}{5}$

metric geometry - area

Question:
If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is

<image1>
(A) $\frac{1}{3}$
(B) $\frac{2}{5}$
(C) $\frac{7}{15}$
(D) $\frac{8}{15}$
(E) $\frac{3}{5}$


Question:
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.

<image1>

Answer: 260

metric geometry - length

Question:
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.

<image1>

Answer: 260