Section A
Multiple Choice Questions — 15 Questions
(+4 / -1)
1
number-theory
easy
Find the possible value(s) of √(√0.0081 + √0.0001 + √0.81).
The source PDF prints no A/B/C/D/E letters for these options; labels here follow the exact on-page top-to-bottom order.
A
1
B
-1
C
Both 1 and -1
D
Decimal numbers do not have square roots
E
√0.019
Show answer
Answer: A
Taking the principal (non-negative) square root of each decimal: √0.0081 = 0.09, √0.0001 = 0.01, and √0.81 = 0.9. Adding these gives 0.09 + 0.01 + 0.9 = 1. Taking the square root of that sum: √1 = 1.
2
fractions
easy
If 5 is added to the numerator and denominator of the fraction 6/11, how will the value of the fraction change?
A
No change
B
Increase
C
Decrease
D
It increases by 11/16
E
The fraction evaluates to 1
Show answer
Answer: B
Adding 5 to both parts gives a new numerator of 6 + 5 = 11 and a new denominator of 11 + 5 = 16, so the new fraction is 11/16. (Option D confuses the amount added to the numerator/denominator with the change in the value of the fraction itself.) Converting both fractions to decimals to compare: 6/11 ≈ 0.545 and 11/16 ≈ 0.688. Since 0.688 > 0.545, the fraction increases in value.
3
rate-problems
medium
If a mother makes x cakes in 2 hours and y pizzas in 3 hours, how many cakes and pizzas altogether can she make in 6 hours? Assume she is making both together and that she works just as fast as if she made them separately.
A
6(x/2 + y/3)
B
6(y/2 + x/3)
C
6(2x + 3y)
D
6(3x + 2y)
E
30
Show answer
Answer: A
Use the unitary method: figure out the rate per hour, then scale up to 6 hours. In one hour she makes, on average, x/2 cakes and y/3 pizzas. In 6 hours, she therefore makes 6(x/2 + y/3) items in total. (In the real world you might save time by doing preparatory steps for both dishes at once, but the problem tells us to assume she works just as fast whether making them together or separately, so no such time saving applies here.)
4
ratio-proportion
medium
Sasha wants to fit an image into a box on her computer screen. Her original image was 1½ inches wide and 2¼ inches tall. Her computer screen box is 4 inches wide. She needs to adjust the box on her computer screen. How tall does the computer screen box need to be for her to fit the image exactly and snugly into the box, without squishing or stretching the image? The choices shown are in inches.
A
9/32
B
27/32
C
6
D
4 3/4
E
2 1/2
Show answer
Answer: C
To keep an image from looking squished or stretched, both dimensions must be scaled by the same factor (stretching/shrinking is a matter of multiplication/division, not addition/subtraction). The original width is 1½ inches, and the new width is 4 inches, so the scale factor is 4 ÷ 1½ = 4 ÷ 3/2 = 4 × 2/3 = 8/3. Applying the same factor to the height: 2¼ × 8/3 = 9/4 × 8/3 = 6 inches.
5
algebra
medium
I have a set of four numbers P, Q, R, S in ascending order. The average of P, Q, and R is 22, and the average of Q, R, and S is 24. The largest number is 27. Then, the smallest number in this set:
A
Is prime
B
Is multiple of 9
C
Is multiple of 11
D
Is multiple of 7
E
Can not be determined with this information
Show answer
Answer: D
From the first average, (P+Q+R)/3 = 22, so P+Q+R = 66 (Equation 1). From the second average, (Q+R+S)/3 = 24, so Q+R+S = 72. Since P, Q, R, S are in ascending order, S is the largest number, so S = 27. Then Q+R = 72 − 27 = 45. Substituting into Equation 1: P + 45 = 66, so P = 21. Since P, Q, R, S are in ascending order, P is the smallest number in the set. 21 = 3 × 7, so P is a multiple of 7 (it is not prime, not a multiple of 9, and not a multiple of 11).
6
place-value
easy
The difference in the place values of the numeral 7 in the largest and the smallest numbers formed using 3, 4, 5, 6, 7 (each digit is used once and exactly once) is
A
69993
B
41976
C
76536
D
9999
E
70
Show answer
Answer: A
Arranging the digits 3, 4, 5, 6, 7 in descending order gives the largest number, 76543; arranging them in ascending order gives the smallest number, 34567. In 76543, the digit 7 is in the ten-thousands place, so its place value is 70,000. In 34567, the digit 7 is in the ones place, so its place value is 7. The difference is 70,000 − 7 = 69,993.
7
fractions
medium
Assume that all sections within each of the given shapes are the same size. Consider only the area represented by the flower, the circle, etc. without including any surrounding area around it. Which fraction below represents the fraction of the shaded part of any shape in the simplest form?
The source PDF prints no A/B/C/D/E letters for these options; labels here follow the exact on-page top-to-bottom order.
A
6/13
B
4/10
C
1/4
D
2/3
E
1/2
Show answer
Answer: D
Working out the shaded fraction of each shape in the figure: shape i has 8 of its 12 equal petals shaded, i.e. 8/12 = 2/3 in simplest form. Shape ii has 2 of 5 sectors shaded, i.e. 2/5 (this is numerically the same as option B's 4/10, but 4/10 is not written in simplest form, so it isn't the shape the question is pointing to). Shape iii has 1 of 3 equal strips shaded, i.e. 1/3. Shape iv has 6 of 11 sectors shaded, i.e. 6/11. Shape v has 7 of 9 petals shaded, i.e. 7/9. Comparing all five shaded fractions (in simplest form) against the five printed options, only 2/3 (shape i) is an exact match, so the answer is 2/3.
8
geometry
medium
A tessellation is a set of figures that can cover a plane without overlapping or leaving any gaps. Tessellations with triangles is shown below. You can continue in the horizontal and vertical directions infinitely to cover a plane. Consider three identical regular pentagons. In how many ways can 3 such pentagons be arranged so that they form a tessellation?
A
Zero
B
One
C
Two
D
Four
E
Infinite
Show answer
Answer: A
A shape can only tessellate the plane on its own if its interior angle is an exact divisor of 360°. A regular triangle's interior angle is 60° (360°/60° = 6), a square's is 90° (360°/90° = 4), and a regular hexagon's is 120° (360°/120° = 3) — all exact divisors. A regular pentagon's interior angle is 108°, and 360°/108° is not a whole number, so no arrangement of regular pentagons (whether 3 or any other number) can tile the plane without gaps or overlaps. There are zero ways.
9
graphs
easy
Look at the following graph. Which of the following sections of the graph have the highest rate of change?
A
A to B
B
B to C
C
A to C
D
D to E
E
E to F
Show answer
Answer: E
The rate of change on a distance-time graph is its slope (gradient) — how steep the line is. Reading the marked points from the graph: A(0,0), B(20,5), C(45,15), D(70,30), E(100,40), F(120,58). The slope of each segment is (change in distance)/(change in time): A to B ≈ 5/20 = 0.25 km/min, B to C ≈ 10/25 = 0.4 km/min, D to E ≈ 10/30 ≈ 0.33 km/min, and E to F = 18/20 = 0.9 km/min. Even though the segments span different lengths on the time axis, the E to F segment is visibly the steepest, and has the highest computed slope. So the rate of change is highest from E to F.
10
spatial-reasoning
medium
A paper is folded as shown below and punched in its folded state. When the paper is opened out again, what will be the pattern of holes formed?
The source PDF prints no A/B/C/D/E letters for these options; labels here follow the exact on-page top-to-bottom order. Each option below is itself an image; the correct option was determined by visually matching the small confirmation image printed in the paper's own Answers section against these four option images.
Show answer
Answer: D
The source's own suggested approach for this question is to physically test it: 'Cut out a piece of paper and try it out.' Folding the triangle along the marked lines shown in the three fold-step diagrams and punching a hole through all the folded layers reproduces, once the paper is unfolded, a pattern with three dots clustered together near the left side of the triangle and a fourth dot near the bottom-right corner — this matches option D exactly, including matching the paper's own printed confirmation image for this question.
11
sets
medium
Which of the following Venn diagrams represents soccer players, high school Math enthusiasts, and students?
The source PDF prints no A/B/C/D/E letters for these options; labels here follow the exact on-page top-to-bottom order. Each option below is itself an image.
Show answer
Answer: B
Analyze what is known about each group: every high-school Math enthusiast is necessarily a student (so the 'Math' circle must sit entirely inside the 'Students' circle); soccer players may or may not be students, and may or may not be Math enthusiasts (so the 'Soccer' circle must overlap the Students/Math region while also extending outside it); and students in general may or may not be Math enthusiasts or soccer players. This calls for three differently-sized, overlapping circles: a large 'Students' circle, a medium 'Math' circle mostly overlapping/nested inside it, and a smaller 'Soccer' circle overlapping the right-hand side of both — which is exactly option B, and matches the paper's own hand-drawn Students/Math/Soccer diagram in its Solutions section.
12
sets
easy
40 people are in a movie theatre which offers two snack options – popcorn and chips. 26 people like popcorn but they do not like chips. 32 people like popcorn. Everyone in the theatre likes at least one of the two snack options. Find the number of people who like both.
A
6
B
8
C
32
D
40
E
Indeterminate
Show answer
Answer: A
Of the 32 people who like popcorn, 26 like popcorn only (not chips). The remaining popcorn-likers must also like chips, so the number who like both is 32 − 26 = 6.
13
geometry
medium
The surface areas of the two spheres are in the ratio 4:9. Find the ratio of their volumes.
A
2:3
B
16:31
C
3:2
D
4:9
E
8:27
Show answer
Answer: E
Surface area of a sphere is 4πr², so the surface-area ratio 4πr²/4πR² = 4/9 gives r²/R² = 4/9, so r:R = 2:3. Volume of a sphere is (4/3)πr³, so the volume ratio is r³/R³ = 2³/3³ = 8/27. The volumes are in the ratio 8:27.
14
number-theory
easy
How many three-digit natural numbers are divisible by 7?
A
141
B
142
C
143
D
128
E
129
Show answer
Answer: D
Three-digit numbers run from 100 to 999. Using the arithmetic progression formula aₙ = a + (n−1)d with the last three-digit multiple of 7 before 1000, aₙ = 994, a = 7 (or the first three-digit multiple, 105 — either works with the appropriate a and count), d = 7: substituting a = 7 (i.e. counting all multiples of 7 up to 994) gives 994 = 7 + (n−1)(7), so n = 142. But we only want the three-digit ones, so we subtract the 14 multiples of 7 that are one- or two-digit (7×1 through 7×14=98): 142 − 14 = 128.
15
coordinate-geometry
easy
A lattice point is a point in a Cartesian coordinate system such that both its x and y-coordinates are integers. How many lattice points are there in a line with (2,4), (3,6)?
A
12
B
6
C
1
D
2
E
Infinitely many
Show answer
Answer: E
The slope between (2,4) and (3,6) is (6−4)/(3−2) = 2, so the line's equation is y − 4 = 2(x − 2), i.e. y = 2x. For every integer value of x, y = 2x is also an integer, so this line passes through infinitely many lattice points.
Section B
Open-Ended (Integer) Questions — 10 Questions
(+5)
16
number-theory
hard
Suppose you have a number with 2021 digits. Let P be its sum of digits. Let Q be the sum of the digits of P. Let R be the sum of the digits of Q. (Assume simple addition but not digital sum or recursive sum.) Find the largest value of Q.
Show answer
Answer: 35
The maximum possible digit sum of a 2021-digit number is 9 × 2021 = 18,189 (every digit equal to 9), so P can be any integer from 1 up to 18,189. To make Q = digit-sum(P) as large as possible, look for the value of P (within that range) whose own digits sum to the most. Trying P = 17,999 (a 5-digit value close to the maximum, with the largest possible trailing digits): Q = 1+7+9+9+9 = 35.
17
combinatorics
hard
Given
X Y Z
+ Z Y X
----------
A B A A
where A, B, X, Y, Z are unique positive single-digit numbers, what is the number of possible combinations for X and Z to satisfy this equation?
Show answer
Answer: 6
Since each letter is a single digit, any carry between columns is at most 1. The sum is a 4-digit number while both addends are 3-digit, so the thousands digit A must equal the carry out of the hundreds column, and so A = 1. In the ones column, X + Z must produce a ones digit of A = 1; since X and Z are positive digits, X + Z = 1 is too small, so there must be a carry: X + Z = 11 (carrying 1 into the tens column). In the tens column, Y + Y + 1 must produce a tens digit of A = 1 with a carry into the hundreds column, so 2Y + 1 = 1 + 10, giving Y = 5. Since X and Z sum to 11 and Y = 5, neither X nor Z can equal 5. The digit pairs (X,Z) summing to 11 are then {(2,9),(3,8),(4,7)} or their reverses {(9,2),(8,3),(7,4)} — 6 possibilities in total. Checking: 259+952 = 1211, 358+853 = 1211, 457+754 = 1211, and their reverses, all fit the ABAA pattern with A=1.
18
3d-geometry
hard
A steel cuboid is reshaped into a cube. Initially, its length, breadth, and depth were 270 cm, 100 cm, and 64 cm respectively. Find the sum of digits of the surface area of the cube.
Show answer
Answer: 18
Reshaping preserves volume (the amount of material used stays the same), so the cube's volume equals the cuboid's volume: 270 × 100 × 64 = 1,728,000 cm³. The side of the cube is the cube root of 1,728,000. Since this number ends in three zeroes, the cube's side ends in exactly one zero. Looking at 1728 (ignoring the trailing zeroes): it ends in 8, so the side's tens digit is 2 (since only a number ending in 2 has a cube ending in 8); and 1728 is the cube of a two-digit number between 10 and 20, which must be 12 (12³ = 1728). So each side of the cube is 120 cm. Total surface area of a cube = 6 × side × side = 6 × 120 × 120 = 86,400 cm². Sum of digits = 8+6+4+0+0 = 18.
19
ratio-proportion
medium
If (3x−4):(y+15) is a fixed ratio and x = 7/3, then y = 12. Find the value of x + y when x = 2.
Show answer
Answer: 5
When x = 7/3, the first term becomes 3×(7/3) − 4 = 7 − 4 = 3, and the second term becomes 12 + 15 = 27, so the fixed ratio is 3:27 = 1:9. When x = 2, the first term becomes 3×2 − 4 = 6 − 4 = 2. Using the fixed ratio 1:9 (i.e. 2:? = 3:27, which simplifies the same way), the second term must be 2 × 9 = 18, so y + 15 = 18, giving y = 3. Cross-check: with x=2 and y=3, the ratio of the two terms is 2:18 = 1:9, matching the fixed ratio. Therefore, x + y = 2 + 3 = 5.
20
number-theory
easy
What is the sum of prime numbers between 2000 and 2021?
Show answer
Answer: 6031
Checking each number strictly between 2000 and 2021 for primality: 2003, 2011, and 2017 are prime (2009 = 7×287, 2013 = 3×671, 2015 = 5×403, and 2019 = 3×673 are all composite; 2021 = 43×47 is composite and also excluded as an endpoint). Their sum is 2003 + 2011 + 2017 = 6031.
21
coordinate-geometry
medium
A lattice point is a point in a Cartesian coordinate system such that both its x and y-coordinates are integers. A point lattice is constructed by plotting all of the points (a,b) such that a and b are positive integers. How many points in the point lattice lie on the line y = −4x + 8?
Show answer
Answer: 1
The line is y = −4x + 8. Since both a and b must be positive integers, we need y > 0, i.e. −4x + 8 > 0, so x < 2. Trying positive integer values: when x = 1, y = 4 (a valid point, (1,4)); when x = 2, y = 0, which is discarded since y must be a positive integer, not zero. No smaller positive integer x is available. So there is only 1 point in the point lattice on this line.
22
data-interpretation
medium
Several people were surveyed for their preferences for 4 drinks A, B, C and D. All the people surveyed are represented in the graph below. If the fraction of people who preferred A is m/n, find m+n.
Show answer
Answer: 16
From the bar graph, the number of people who preferred each drink is A = 150, B = 100, C = 150, and D = 250. The total number of people surveyed is 150 + 100 + 150 + 250 = 650. The fraction who preferred A is 150/650 = 15/65 = 3/13. So m = 3 and n = 13, and m + n = 3 + 13 = 16.
23
number-theory
medium
How many three-digit natural numbers 'n' can there be such that all the three conditions given below are satisfied: Condition 1: n − 14 is divisible by 7, Condition 2: n − 24 is divisible by 8, and Condition 3: n − 36 is divisible by 9.
Show answer
Answer: 1
Since 14 is itself divisible by 7, "n − 14 divisible by 7" is equivalent to "n divisible by 7". Likewise 24 is divisible by 8, so condition 2 is equivalent to "n divisible by 8"; and 36 is divisible by 9, so condition 3 is equivalent to "n divisible by 9". So n must be divisible by 7, 8, and 9 simultaneously — that is, by their LCM. Since 7, 8, and 9 are pairwise coprime, the LCM is 7×8×9 = 504. The only three-digit multiple of 504 is 504 itself (504×2 = 1008, which has four digits). So there is only 1 such number.
24
algebra
hard
Simplify and round off to the nearest whole number: 16/(√14+√10) − 58/(3√5−4) + √180 + √160
Show answer
Answer: 7
This expression can be simplified using conjugate tricks, but it is simplest to verify numerically. √14 ≈ 3.7417 and √10 ≈ 3.1623, so √14+√10 ≈ 6.9039, and 16/6.9039 ≈ 2.317. Next, 3√5 ≈ 6.7082, so 3√5−4 ≈ 2.7082, and 58/2.7082 ≈ 21.411. Also, √180 ≈ 13.416 and √160 ≈ 12.649. Putting it together: 2.317 − 21.411 + 13.416 + 12.649 ≈ 6.972, which rounds to 7.
25
number-theory
hard
Using all 9 digits exactly once - 1, 2, 3, 4, 5, 6, 7, 8, 9, as numerator and denominator, create a fraction equalling 1/3. The digits in the numerator and denominator should together have all the 9 digits shown here. What is the numerator?
Show answer
Answer: 5832
We need a numerator N and denominator D with D = 3N, such that the digits of N and D together use each of 1 through 9 exactly once. Taking N = 5832: 5832 × 3 = 17,496. The numerator's digits are {5,8,3,2} and the denominator's digits are {1,7,4,9,6}; together these are exactly {1,2,3,4,5,6,7,8,9}, each used once. Since 5832/17496 = 1/3, the numerator is 5832.