Section A
Multiple Choice Questions — 15 Questions
(+3)
1
area-counting
easy
A shape is drawn on centimeter grid paper. What is its area in square centimeters? Each small square has area 1 square centimeter.
Answer: C
Count the shaded squares.
The upright part is 2 squares wide and 4 squares tall: 2 × 4 = 8.
The bottom part is 5 squares wide and 2 squares tall: 5 × 2 = 10.
Area = 8 + 10 = 18 square centimeters.
2
number-line
easy
What number is exactly halfway between 5 and 18 on the number line?
Answer: C
The distance from 5 to 18 is 18 − 5 = 13.
Half of 13 is 6 1/2, so the halfway number is 5 + 6 1/2 = 11 1/2.
3
triangles
medium
A copper wire is 24 centimeters long and has centimeter marks on it. Mia bends the wire at two marks to make a triangle. One side is 7 centimeters long, and all side lengths are whole numbers. Two triangles are considered the same if they have the same three side lengths. How many different triangles are possible?
Answer: B
The other two sides add up to 24 − 7 = 17 cm.
In a triangle, the two shorter sides together must be longer than the third side, so the difference of these two sides must be less than 7.
Pairs adding to 17: (8, 9), (7, 10), (6, 11) work; (5, 12) has difference 7 and lies flat, and the others are worse.
So 3 different triangles are possible.
4
spatial-reasoning
medium
A cube net has six letters written on its faces. When the net is folded into a cube, which letter is on the face opposite A?
Answer: D
Keep A on the bottom and fold up the faces around it.
N (above A) and E (right of A) become two side walls. R folds down from N to become another side, and U folds from E to become the side opposite N.
T is attached to U and folds over to cover the top, so T is opposite A.
5
distance
easy
Four towns are on one straight road in the order shown. Town D is 18 miles from Town A. Town C is 11 miles from Town A. Town B is 13 miles from Town D. How far is Town B from Town C?
Answer: B
Town A to Town B = 18 − 13 = 5 miles.
Town B to Town C = 11 − 5 = 6 miles.
6
fractions
easy
Which of the following numbers has the greatest value?
3/4, 5/8, 7/10, 4/5
Answer: D
Write each as a decimal: 3/4 = 0.75, 5/8 = 0.625, 7/10 = 0.7, 4/5 = 0.8.
The greatest is 4/5.
7
geometry-area
easy
A shaded right triangle is drawn on a square grid, as shown below. Each small square is 1 centimeter by 1 centimeter. What is the area of the shaded triangle?
Answer: A
The two sides that meet at the right angle A are 4 cm (A to B) and 6 cm (A to C).
The triangle is half of a 4 by 6 rectangle: 4 × 6 ÷ 2 = 12 square centimeters.
8
time
easy
Noah began a drawing project at 9:35 a.m. He worked for 95 minutes. What time did he finish?
Answer: C
95 minutes = 1 hour 35 minutes.
9:35 a.m. + 1 hour = 10:35 a.m.; 10:35 a.m. + 35 minutes = 11:10 a.m.
9
factors
easy
How many whole numbers divide evenly into 36? Be sure to include 1 and 36.
Answer: B
List the factor pairs of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
The factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36, which is 9 numbers.
10
data-interpretation
easy
A survey was given to 160 students about their favorite season. Any student could select only one season. All students replied. According to the chart, how many students chose spring?
Answer: B
Winter, summer and fall together are 50% + 25% + 10% = 85%.
Spring is 100% − 85% = 15%.
15% of 160 = 160 ÷ 20 × 3 = 24 students.
11
age-problems
medium
In 2026, Lina is 9 years old. Her aunt is 30 years older than Lina. In what year will her aunt be 4 times as old as Lina?
Answer: A
In 2026 the aunt is 9 + 30 = 39.
The difference of 30 years never changes. When the aunt is 4 times as old, the difference is 3 times Lina's age, so Lina is 30 ÷ 3 = 10 and the aunt is 40.
Lina is 10 one year after 2026, in 2027.
12
symmetry
easy
A regular hexagon is shown below. In how many different ways can it be folded in half so that the two halves match perfectly?
Answer: B
Each fold must be along a line of symmetry.
A regular hexagon has 3 lines through opposite corners and 3 lines through the middles of opposite sides, so 6 ways.
13
averages
easy
The average of four numbers is 18. When a fifth number is added, the average becomes 20. What is the fifth number?
Answer: C
Total of the four numbers = 4 × 18 = 72.
Total of the five numbers = 5 × 20 = 100.
The fifth number = 100 − 72 = 28.
14
combinations
easy
The student council is designing a flag for an event. Each flag must have exactly one background color, one border color, and one symbol in the middle. There are 4 choices for the background color, 3 choices for the border color, and 2 choices for the symbol. How many different flags can be designed?
Answer: C
Each background can go with each border, and each of those with each symbol.
Number of flags = 4 × 3 × 2 = 24.
15
series
easy
What is the value of the following sum of whole numbers?
2 + 4 + 6 + 8 + 10 + ⋯ + 38 + 40
Answer: D
There are 20 numbers. Pair them from the outside in: 2 + 40 = 42, 4 + 38 = 42, and so on, giving 10 pairs.
Sum = 10 × 42 = 420.
Section B
Open-Ended Questions — 5 Questions
(+5)
16
patterns
medium
A growing pattern is made from square tiles, as shown in the figure below. If the pattern continues, how many tiles are in Step 100?
Answer: 400
Step n is the border of a square that is (n + 1) tiles on each side.
The tiles are 4, 8, 12, 16, … so Step n has 4 × n tiles.
Step 100 has 4 × 100 = 400 tiles.
17
counting-figures
medium
A 3 by 3 square grid is divided into 9 small squares. How many rectangles of all sizes can be found in the grid?
Answer: 36
A rectangle is fixed by choosing 2 of the 4 vertical lines and 2 of the 4 horizontal lines.
There are 6 ways to choose 2 of 4 lines, so the number of rectangles is 6 × 6 = 36.
(The squares count as rectangles too.)
18
place-value
hard
The number 99 can be written using groups of 4:
99 = A × (4 × 4 × 4) + B × (4 × 4) + C × 4 + D,
where A, B, C, and D are digits chosen from {0, 1, 2, 3}. What is the four-digit number ABCD?
Answer: 1203
4 × 4 × 4 = 64 goes into 99 once, leaving 35, so A = 1.
4 × 4 = 16 goes into 35 twice, leaving 3, so B = 2.
4 goes into 3 zero times, so C = 0, and D = 3.
Check: 64 + 32 + 0 + 3 = 99. ABCD = 1203.
19
area-counting
medium
A board has 100 small square tiles. Some white tiles are shown below. All the other tiles are hidden. How many hidden tiles are there?
Answer: 66
The white tiles form three rectangles: 4 × 4 = 16, 3 × 4 = 12 and 2 × 3 = 6.
White tiles = 16 + 12 + 6 = 34.
Hidden tiles = 100 − 34 = 66.
20
cryptarithm
hard
In the multiplication below, A, B, and C are different digits from {1, 2, 3, 4, 5, 6, 7, 8, 9}. What is the value of (A + B + C)?
Answer: 22
The ones digit of B × A must be B, with A and B different and not 1.
That happens only when A = 6 and B is even, or B = 5 and A is odd.
A = 6: 62 × 6 = 372, 64 × 6 = 384, 68 × 6 = 408 — none has the form CBB.
B = 5: 35 × 3 = 105, 75 × 7 = 525, 95 × 9 = 855 — only 855 fits, so A = 9, B = 5, C = 8.
A + B + C = 9 + 5 + 8 = 22.
Section C
Open-Ended Questions — 5 Questions
(+6)
21
algebra
hard
In the diagram, the numbers A, B, and C are placed in the circles at the corners of the triangle. Each square shows the product of the two numbers in the neighboring circles. What is the value of (A × B × C)?
Answer: 18
Multiply the three products: (A × C) × (B × C) × (A × B) = 12 × 3 × 9 = 324.
This is (A × B × C) × (A × B × C), and 18 × 18 = 324.
So A × B × C = 18.
22
geometry-perimeter
hard
In the figure below, ABCD is a square, and the shaded region FGHE is a rectangle. Points F and G are on side BC, and points E and H are on side AD. If CF = 22 inches and AH = 14 inches, what is the perimeter of rectangle FGHE in inches? Please enter a whole number without units.
Answer: 72
Let BF = AE = x. The side of the square is BF + FC = x + 22, which is also the length FE of the rectangle.
The width EH = AH − AE = 14 − x.
Length + width = (x + 22) + (14 − x) = 36, so the perimeter is 2 × 36 = 72 inches.
23
geometry-area
hard
On a geoboard, all dots are one inch apart in both horizontal and vertical directions. What is the area, in square inches, of the shaded tilted square? Please enter a whole number without units.
Answer: 8
Draw the square's two diagonals: each joins dots 4 inches apart.
The square splits into 4 right triangles, each with legs 2 and 2, area 2 × 2 ÷ 2 = 2.
Area = 4 × 2 = 8 square inches. (Or: the enclosing 4 × 4 square is 16, minus 4 corner triangles of 2 each.)
24
linear-equations
medium
At a school store, there is no tax. Emma bought 2 pencils and 1 eraser for 7 dollars. Leo bought 5 pencils and 1 eraser for 13 dollars. Both students bought the same kinds of pencils and erasers. What is the price of one eraser in dollars? Please enter a whole number without units.
Answer: 3
Leo bought 3 more pencils than Emma and paid 13 − 7 = 6 dollars more, so one pencil costs 2 dollars.
Emma: 2 × 2 + eraser = 7, so the eraser costs 3 dollars.
25
geometry-area
hard
A small triangle has area 10 square centimeters. The triangle grows by adding 3 matching copies around it to make the next larger triangle. It grows this way 3 times, as shown below. What is the area of the large triangle ABC in square centimeters? Please enter a number without units.
Answer: 640
Each growth step makes a triangle out of 4 copies of the previous one, so the area is multiplied by 4.
10 → 40 → 160 → 640.
The area of triangle ABC is 640 square centimeters.