Section A
Multiple Choice Questions — 15 Questions
(+3)
1
geometry-perimeter
easy
An L-shape is placed on top of a triangular ruler, as shown below. Use the ruler markings to help you. What is the perimeter of the L-shape in centimeters?
Answer: C
Following the boundary gives lengths 5, 1, 4, 1, 1, and 2 centimeters. Their sum is 5 + 1 + 4 + 1 + 1 + 2 = 14 centimeters.
2
number-partitioning
medium
Jada has a ribbon that is 30 centimeters long. She cuts it into three pieces: a short piece, a medium piece, and a long piece. The short piece is 4 centimeters. The medium piece is longer than the short piece. The long piece is longer than the medium piece. All pieces are in whole numbers. What is the longest possible length of the long piece?
Answer: C
To maximize the long piece, make the medium piece the smallest whole number longer than 4, which is 5. The long piece is then 30 − 4 − 5 = 21 centimeters.
3
number-line
easy
There is a number that is halfway between 2 and 14 on the number line. What is the value of that number?
Answer: D
The halfway value is the average: (2 + 14) ÷ 2 = 8.
4
spatial-reasoning
medium
A cube template has six letters written on its faces. When the cube is folded, which letter is on the face opposite the letter A?
Answer: B
When the net is folded, M and T become opposite faces, E and Z become opposite faces, and the extra face H folds around to lie opposite A.
5
distance
medium
There are four houses along a country road: House A, House B, House C, and House D, shown below. House D is 10 miles away from House A. House C is 5 miles away from House A. House D is 7 miles away from House B. How far is House C from House B?
Answer: B
Place A at 0 and D at 10. Since D is 7 miles from B, B is at 3. Since C is 5 miles from A, C is at 5. Therefore BC = 5 − 3 = 2 miles.
6
fractions
easy
Which of the following numbers has the greatest value?
Answer: A
The fractions are approximately 0.83, 0.40, 0.67, and 0.60. Therefore 5/6 is the greatest.
7
time
easy
Tim started reading a book at 8:50 a.m. He read for 75 minutes. What time did he finish reading?
Answer: C
Seventy-five minutes is 1 hour 15 minutes. Adding this to 8:50 a.m. gives 10:05 a.m.
8
factors
easy
How many whole numbers divide evenly into 24? Be sure to include 1 and 24.
Answer: A
Since 24 = 2³ × 3¹, the number of positive divisors is (3 + 1)(1 + 1) = 8. They are 1, 2, 3, 4, 6, 8, 12, and 24.
9
geometry-area
medium
A pinwheel shape is drawn on graph paper as shown below. What is the total area of the pinwheel in square centimeters?
Answer: C
The blue cross has area 12 square centimeters. Each of the four pink triangles has base 2 and height 2, so each has area 2. The total is 12 + 4 × 2 = 20 square centimeters.
10
fractions
medium
A survey was given to 120 students about their favorite fruit. One student could select only one fruit. All students replied. A pie chart was made of the data. How many students like oranges according to the chart?
Answer: C
The orange fraction is 1 − 1/2 − 1/4 − 3/20 = 1/10. Therefore 120 × 1/10 = 12 students like oranges.
11
age-problems
easy
Linda is 8 years old in 2025. Her cousin is 10 years older than she is. In what year will her cousin be twice as old as Linda?
Answer: B
Linda's cousin is 18. After x years, 18 + x = 2(8 + x), so x = 2. Two years after 2025 is 2027.
12
symmetry
medium
Look at the regular octagon below. In how many different ways can it be folded in half so that the two halves match perfectly?
Answer: A
A regular octagon has 8 lines of symmetry: 4 through pairs of opposite vertices and 4 through the midpoints of pairs of opposite sides.
13
averages
easy
The average of three numbers is 15. When a fourth number is added, the average becomes 18. What is the value of the fourth number?
Answer: B
The first three numbers total 3 × 15 = 45. All four total 4 × 18 = 72. The fourth number is 72 − 45 = 27.
14
combinations
easy
Lily has three shirts (blue, green, pink) and five pairs of pants (black, blue, gray, green, and red). How many different outfits can she make by choosing one shirt and one pair of pants?
Answer: B
Each of the 3 shirts can be paired with any of the 5 pairs of pants, giving 3 × 5 = 15 outfits.
15
geometry-perimeter
hard
In the following figure, ABCD is a square; the shaded region EFGH is a rectangle. AH = 14 inches, and FC = 22 inches. What is the perimeter of the shaded rectangle EFGH in inches?
Answer: D
Let the shaded width be w and the left and right margins be x and y. Then AH = x + w = 14 and FC = y + w = 22. The rectangle's perimeter is 2[(x + w + y) + w] = 2[(x + w) + (y + w)] = 2(14 + 22) = 72 inches.
Section B
Open-Ended Questions — 5 Questions
(+5)
16
patterns
easy
Five steps of a growing pattern are shown below. If the pattern goes on, how many marbles will there be in the 10th step?
Answer: 55
The numbers of marbles are the triangular numbers 1, 3, 6, 10, 15, and so on. Step n has n(n + 1) ÷ 2 marbles, so step 10 has 10 × 11 ÷ 2 = 55.
17
counting-triangles
hard
A big triangle is divided into smaller triangles. How many triangles of all sizes are there?
Answer: 53
Count by size in the recursive figure: there are 36 smallest triangles, 12 triangles with side length 2 units, 4 triangles with side length 4 units, and the 1 whole triangle. The total is 36 + 12 + 4 + 1 = 53.
18
arithmetic
easy
What is the value of the following number sentence? (15 + 7 + 97 + 997) × 5
Answer: 5580
First add inside the parentheses: 15 + 7 + 97 + 997 = 1116. Then 1116 × 5 = 5580.
19
area-counting
medium
A hundred chart is partially shaded. How many small squares are there in the unshaded (white) region?
Answer: 48
The white region contains a top rectangle of 8 × 4 = 32 squares and a lower rectangle of 4 × 4 = 16 squares. Thus it contains 32 + 16 = 48 squares.
20
digit-algebra
hard
In the following problem, A and B are each a different number from {1, 2, 3, 4, 5, 6, 7, 8, 9}. In other words, AB × B = 3BA. What is the value of A × B?
Answer: 32
The multiplication is 48 × 8 = 384, so A = 4 and B = 8. Therefore A × B = 4 × 8 = 32.
Section C
Open-Ended Questions — 5 Questions
(+6)
21
algebra
hard
In the diagram below, three numbers A, B, and C are placed at the corners of the triangle. Each square has the product of the two numbers in the neighboring circles. For example, B × C = 20. What is the value of A × B × C?
Answer: 60
The three products are AB = 12, AC = 15, and BC = 20. Their product is (ABC)² = 12 × 15 × 20 = 3600. Therefore ABC = 60.
22
series
medium
What is the value of the following sum of whole numbers? 1 + 3 + 5 + 7 + 9 + 11 + · · · + 23 + 25 + 27 + 29
Answer: 225
This is the sum of the first 15 odd numbers. The sum of the first n odd numbers is n², so the result is 15² = 225.
23
geometry-area
hard
On a geoboard, all dots are one inch apart in both horizontal and vertical directions. Two squares are marked on the geoboard. What is the area, in square inches, of the unshaded (white) region? Please enter a number without units.
Answer: 17
A side of the tilted white square moves 4 units horizontally and 1 unit vertically. Its area is therefore 4² + 1² = 17 square inches.
24
linear-equations
medium
At a local bookstore, there is no tax on school supplies. Josh bought 1 notebook and 2 toy cars for 14 dollars. Tony bought 1 notebook and 1 toy car for 11 dollars. Both boys bought the same kinds of notebooks and toy cars. What is the price of one notebook? Please enter a number without units.
Answer: 8
Subtracting Tony's purchase from Josh's shows that one toy car costs 14 − 11 = 3 dollars. Therefore one notebook costs 11 − 3 = 8 dollars.
25
geometry-area
hard
Triangle ABC is divided again and again, following the pattern below. All the points on the triangle sides are midpoints. The small shaded triangle has an area of 5 square centimeters. What is the area of the big triangle ABC in square centimeters? Please enter a number without units.
Answer: 320
At each midpoint division, the central triangle has 1/4 the area of its parent triangle. The shaded triangle is reached after three such divisions, so its area is 1/4³ = 1/64 of triangle ABC. Therefore the area of ABC is 5 × 64 = 320 square centimeters.