Section A
Multiple Choice Questions — 15 Questions
(+3)
1
counting-triangles
medium
How many triangles are there in the figure below?
Answer: D
There are 6 smallest triangles, 2 medium triangles with vertex at the top and base along the middle horizontal line, the central inverted triangle, 2 larger half-triangles, 2 triangles formed by the top vertex and the central diagonals, and the whole outer triangle. The total is 6 + 2 + 1 + 2 + 2 = 13.
2
combinations
easy
There were five soccer teams. Each soccer team played a match with every other team. How many matches were played?
Answer: C
Each match is determined by choosing 2 of the 5 teams. The number of pairs is 5 × 4 ÷ 2 = 10.
3
counting-squares
easy
In the figure below, all the small squares are unit squares. How many squares of any size are there in the 3 × 3 grid?
Answer: C
There are 9 squares of size 1 × 1, 4 squares of size 2 × 2, and 1 square of size 3 × 3. Thus there are 9 + 4 + 1 = 14 squares.
4
factors
medium
The factor tree of 2023 is given below. How many whole numbers are there that divide evenly into 2023, including 1 and 2023?
Answer: C
The factor tree gives 2023 = 7 × 17 × 17 = 7¹ × 17². A divisor can contain 7 to the power 0 or 1 and 17 to the power 0, 1, or 2, giving (1 + 1)(2 + 1) = 6 divisors.
5
distance
medium
There are four houses located along a curved road. Jason biked from A to D and found that the distance is 12 miles. Katie biked from A to C and found that the distance is 6 miles. Matt biked from D to B and found that it is 8 miles. What is the distance between B and C in miles?
Answer: B
Let AB = x, BC = y, and CD = z. Then x + y + z = 12, x + y = 6, and y + z = 8. The first two equations give z = 6, so y + 6 = 8 and y = 2 miles.
6
averages
easy
Five students were raising money for a school event. After a week, they found that, on average, they each raised 25 dollars. Then, at the last minute, they received another donation. Now the average went up to 30 dollars per person. How much was the last donation in dollars?
Answer: B
The original total was 5 × 25 = 125 dollars. The new total is 5 × 30 = 150 dollars. The last donation was 150 − 125 = 25 dollars.
7
geometry-area
medium
In the tangram shown below, all the points are the midpoints of their respective segments. The big square has an area of 64 square inches. What is the area of the small parallelogram labeled W?
Answer: B
The midpoint construction divides the tangram square into the standard seven tangram pieces. The parallelogram occupies 1/8 of the whole square, so its area is 64 ÷ 8 = 8 square inches.
8
digit-algebra
hard
AB is a two-digit whole number and B is a single-digit number. AB × B = 3AB, where A and B are both numbers from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. What is A × B?
Answer: A
The multiplication is 75 × 5 = 375, so A = 7 and B = 5. Therefore A × B = 7 × 5 = 35.
9
patterns
medium
If the following pattern continues, how many balls are there in the 10th step?
Answer: B
The steps contain the triangular numbers 1, 3, 6, 10, 15, 21, and so on. Step n contains n(n + 1) ÷ 2 balls. Step 10 therefore contains 10 × 11 ÷ 2 = 55 balls.
10
geometry-area
hard
In the following figure, each dot is one inch away from its horizontal and vertical neighbors. Each small square has an area of one square inch. What is the area of triangle ABC in square inches?
Answer: C
Assign grid coordinates A = (0, 3), B = (3, 0), and C = (4, 4). Using the shoelace formula, the area is |0×0 + 3×4 + 4×3 − 3×3 − 0×4 − 4×0| ÷ 2 = 15 ÷ 2 = 7.5 square inches.
11
rates
easy
John is walking around in the neighborhood. In 45 minutes, he walked 2.7 miles. If he keeps the same pace, how many miles will he have walked in two hours?
Answer: B
Two hours is 120 minutes. At the same pace, John walks 2.7 × 120 ÷ 45 = 2.7 × 8/3 = 7.2 miles.
12
number-patterns
hard
The first and last digit of a 5-digit number is shown in the following figure. If the sum of any three adjacent (or neighboring) digits is 10, what is the sum of all the digits of the number?
Answer: A
Write the digits as 5, b, c, d, 3. From 5 + b + c = 10, b + c = 5. Comparing this with b + c + d = 10 gives d = 5. Then c + 5 + 3 = 10 gives c = 2, so b = 3. The digit sum is 5 + 3 + 2 + 5 + 3 = 18.
13
angles-and-time
medium
A clock reads exactly 9:30, as shown in the figure below. What is the measure of the angle between its hour and minute hands?
Answer: C
At 9:30 the minute hand is at 180°. The hour hand has moved halfway from 9 to 10, so it is at 9.5 × 30° = 285°. The smaller angle is 285° − 180° = 105°.
14
volume
easy
Two boxes (also known as rectangular prisms) are built. The big box is twice as much as the small one in width, length, and height. How many small boxes can fit into the big box?
Answer: C
The big box fits 2 small boxes along each of its three dimensions. Therefore it holds 2 × 2 × 2 = 8 small boxes.
15
parity
medium
A number machine takes two numbers as its inputs: 4 and 6. You can enter as many 4's or 6's as you like. Within the machine, they can be added to or subtracted from each other. For example, you can do 6 − 4 + 6 + 6 − 4. Y is the output of the machine. Which of the following numbers is impossible for the machine to create?
Answer: B
Both 4 and 6 are even. Adding or subtracting any number of even numbers always produces an even result. Therefore the odd number 15 is impossible.
Section B
Open-Ended Questions — 5 Questions
(+5)
16
number-partitioning
medium
A 28-inch-long block was cut into a maximum number of blocks of different sizes. The length of each block is a whole number of inches. What is the length of the largest block?
Answer: 7
To maximize the number of different whole-number lengths, use the smallest possible distinct lengths: 1, 2, 3, 4, 5, 6, and 7. Their sum is 28 exactly, so the largest block is 7 inches long.
17
magic-square
hard
The following is a 3 × 3 square. The sum of the numbers in any row, column or diagonal are the same. What is the value of E?
Answer: 9
Let the common sum be S. From the bottom row, F = S − 13. The middle column gives 7 + D + F = S, so D = 6. The left column gives A = S − 11, and the diagonal through C, D, and 8 gives C = S − 14. The top row then gives (S − 11) + 7 + (S − 14) = S, so S = 18. From the middle row, 3 + 6 + E = 18, hence E = 9.
18
fractions
medium
There are some paper stars in a box. One-half of them are blue, one-sixth of them are orange, and the rest are gray. We know there are 30 gray stars. How many stars are blue?
Answer: 45
The gray fraction is 1 − 1/2 − 1/6 = 1/3. If 1/3 of the stars is 30, there are 90 stars in total. Half of 90 is 45, so there are 45 blue stars.
19
geometry-area
hard
Sarah has three squares, one with side length 4, one with side length 6, and one with side length 8. She arranges them so that the corner of the second square is at the center of the first square, and that the corner of the third square is at the center of the second square. The result is the figure below. What is the area of this figure as a whole?
Answer: 103
The three areas total 4² + 6² + 8² = 16 + 36 + 64 = 116. The first two squares overlap in a 2 × 2 square of area 4. The second and third overlap in a 3 × 3 square of area 9. The first and third do not overlap with positive area. Thus the union has area 116 − 4 − 9 = 103.
20
patterns
medium
Peter is making some patterns to display on his bedroom wall. If he continues the pattern below, how many small triangles will there be in Pattern 9?
Answer: 100
Pattern 1 has 2 small triangles along each side, Pattern 2 has 3, and Pattern 3 has 4. Therefore Pattern n has n + 1 along each side and contains (n + 1)² small triangles. Pattern 9 has 10² = 100 small triangles.
Section C
Open-Ended Questions — 5 Questions
(+6)
21
probability
medium
Mary counted all the marbles in her collection and made a bar graph shown below. The marbles are mixed thoroughly. Her friend, Nancy, took 30 marbles without looking. How many green marbles can Nancy expect to have?
Answer: 10
The graph shows 20 gray, 40 green, 25 blue, and 35 orange marbles, for 120 marbles total. The green fraction is 40/120 = 1/3. Nancy can expect 30 × 1/3 = 10 green marbles.
22
linear-equations
medium
In the following diagram, three numbers A, B, C are placed at the corners of a triangle. Each square has the sum of the two numbers in the nearby circles. For example, A + B = 8. What is the value of C?
Answer: 18
The diagram gives A + B = 8, A + C = 21, and B + C = 23. Adding the last two equations and subtracting the first gives 2C = 21 + 23 − 8 = 36, so C = 18.
23
geometry-area
medium
Equilateral triangle ABC is divided using the midpoints of its sides to form another triangle. The pattern continues for the shaded triangle. If the area of triangle ABC is 64 square centimeters, what is the area of the shaded triangle in square centimeters? Please enter a whole number without the units.
Answer: 4
Joining the midpoints creates a central triangle with 1/4 the area of the original, so its area is 64 ÷ 4 = 16. Repeating the midpoint construction makes the shaded triangle 1/4 of that central triangle, so its area is 16 ÷ 4 = 4.
24
geometry-perimeter
hard
In the following figure, ABCD is a square; the shaded part EFGH is a rectangle. AE = 6 cm, CG = 12 cm. What is the perimeter of the shaded rectangle EFGH in cm?
Answer: 36
Let the left margin be x, the rectangle width be w, and the right margin be y. Then AE = x + w = 6 and CG = y + w = 12. The square's side is x + w + y, so the rectangle's perimeter is 2[(x + w + y) + w] = 2[(x + w) + (y + w)] = 2(6 + 12) = 36 cm.
25
age-problems
medium
Mary is sixteen years old and is four times as old as her brother. How old will Mary be when she is twice as old as her brother?
Answer: 24
Mary's brother is currently 16 ÷ 4 = 4 years old, so their age difference is 12 years. When Mary is twice her brother's age, that 12-year difference equals her brother's age. Her brother will be 12 and Mary will be 24.