Section A
15 Multiple Choice Questions
(+3)
1
Absolute Value Inequalities
easy
Which of the following intervals represents the solution set to the absolute value inequality? |x − 2| ≤ 15
A
(−13, 17)
B
(13, −17)
C
[−13, 17]
D
[−13, −17)
E
(−13, −17]
Show answer
Answer: C
|x − 2| ≤ 15 means −15 ≤ x − 2 ≤ 15. Adding 2 to all parts: −13 ≤ x ≤ 17. Since the inequality is ≤ (not strict), both endpoints are included, giving the closed interval [−13, 17]. Answer: C.
2
Number Patterns
hard
If the following dot pattern continues, how many dots are there at the 25th step?
A
925
B
1001
C
1080
D
1162
E
2022
Show answer
Answer: B
Counting the dots directly from the figure at each step gives 5, 12, 22, 35 dots for Steps 1, 2, 3, 4. The first differences are 7, 10, 13 and the second differences are constant at 3, so the count follows a quadratic in the step number n. Fitting a formula: a(n) = (3n² + 5n + 2)/2 = (3n + 2)(n + 1)/2. Check: a(1) = 3·2/2 = 5, a(2) = 8·3/2 = 12, a(3) = 11·4/2 = 22, a(4) = 14·5/2 = 35 — all match. For n = 25: a(25) = (3·25 + 2)(25 + 1)/2 = 77·26/2 = 77·13 = 1001. Answer: B.
3
Angles at a Point
medium
In the figure below, AB and CD intersect at a point O. m∠DOA = (x + y)°, m∠BOD = (2x − 5y)°, and m∠COB = (3x − 7y)°. What is the value of y?
A
20
B
22.5
C
34
D
36.5
E
40
Show answer
Answer: B
Since A, O, B are collinear, ray OD splits the straight angle AOB into ∠DOA and ∠BOD, so (x + y) + (2x − 5y) = 180, i.e. 3x − 4y = 180. Since C, O, D are collinear, ray OB splits the straight angle COD into ∠COB and ∠BOD, so (3x − 7y) + (2x − 5y) = 180, i.e. 5x − 12y = 180. Multiply the first equation by 3: 9x − 12y = 540. Subtract the second equation: 4x = 360, so x = 90. Substitute back: 3(90) − 4y = 180 → 270 − 4y = 180 → y = 22.5. (Check: ∠DOA = x+y = 112.5°, ∠COB = 3x−7y = 112.5°, matching as vertical angles, confirming the solution.) Answer: B.
4
Geometric Series
medium
April was making seven bracelets for her friends. It took her 5 minutes to make the first bracelet. Each bracelet after the first one took 20% less time than the one created before. How much time, rounded to the nearest minute, did it take April to make all the seven bracelets?
A
11 minutes
B
12 minutes
C
15 minutes
D
20 minutes
E
25 minutes
Show answer
Answer: D
The times form a geometric sequence with first term 5 and common ratio 0.8, for 7 terms. Total = 5·(1 − 0.8⁷)/(1 − 0.8). Computing 0.8⁷ = 0.2097152, so total = 5·(1 − 0.2097152)/0.2 = 5·0.7902848/0.2 = 5·3.951424 = 19.75712 minutes, which rounds to 20 minutes. Answer: D.
5
Least Common Multiple
easy
Paul and Matt ran into each other at a local restaurant today. Paul said he visits the restaurant every 11 days; Matt said he visits the restaurant every 17 days. If they both keep their schedules, in how many days are they going to see each other again at the restaurant?
A
117
B
125
C
28
D
181
E
171
Show answer
Answer: D
They will next meet after a number of days equal to the least common multiple of their visiting cycles. Since 11 and 17 are both prime, LCM(11, 17) = 11 × 17 = 187. This exact value of 187 does not appear among the five printed answer choices (117, 125, 28, 181, 171) — this is a defect in the source paper. Option D (181) is the closest available choice to the correct value of 187 and has been selected as the nominal answer, but note that no option is mathematically correct.
6
Circle Theorems
medium
In the figure below, the circle has its center at C. Points F, G and H are all on the circle. If m∠GFH = 60⁰, what is the measure of ∠GCH in radians?
A
(2/3)π
B
(4/3)π
C
(5/3)π
D
(3/4)π
E
(2/5)π
Show answer
Answer: A
∠GFH is an inscribed angle subtending arc GH (the arc not containing F), and ∠GCH is the central angle subtending the same arc. By the inscribed angle theorem, the central angle is twice the inscribed angle: ∠GCH = 2 × 60° = 120°. Converting to radians: 120° × (π/180°) = (2/3)π. Answer: A.
7
Absolute Value Equations
medium
Let A and B be the two solutions to the following absolute value equation: |−5x + 12| − 8 = 10. What is (A · B)?
A
7.2
B
−7.2
C
8.4
D
−8.4
E
12
Show answer
Answer: B
|−5x + 12| − 8 = 10 gives |−5x + 12| = 18. Case 1: −5x + 12 = 18 → −5x = 6 → x = −1.2. Case 2: −5x + 12 = −18 → −5x = −30 → x = 6. So A = −1.2 and B = 6 (or vice versa). A·B = −1.2 × 6 = −7.2. Answer: B.
8
Linear Equations
easy
The relationship between degrees Fahrenheit (°F) and degrees Celsius (°C) is given in the following equation: 5F = 9C + 160. When a thermometer reads 86°F, what is the temperature in degrees Celsius?
A
25°C
B
26°C
C
27°C
D
30°C
E
32.5°C
Show answer
Answer: D
Substitute F = 86: 5(86) = 9C + 160 → 430 = 9C + 160 → 9C = 270 → C = 30. Answer: D.
9
Systems of Linear Equations
medium
We know that 3x + 5y = 31 and 6x − 2y = 2, where x and y are whole numbers. What is the value of (x·y)?
A
8
B
10
C
12
D
15
E
18
Show answer
Answer: B
From 6x − 2y = 2, divide by 2: 3x − y = 1, so y = 3x − 1. Substitute into 3x + 5y = 31: 3x + 5(3x − 1) = 31 → 3x + 15x − 5 = 31 → 18x = 36 → x = 2. Then y = 3(2) − 1 = 5. Both are whole numbers as required. x·y = 2 × 5 = 10. Answer: B.
10
Similar Triangles / Area Ratios
hard
In the following figure, D is the midpoint between A and C; F is the midpoint between B and C; E is the midpoint between D and C; G is the midpoint between F and C. What is the ratio between the area of ΔABC and the area of ΔEGC?
A
2:1
B
3:1
C
4:1
D
8:1
E
16:1
Show answer
Answer: E
Since D is the midpoint of AC, CD = (1/2)CA; since E is the midpoint of DC, CE = (1/2)CD = (1/4)CA. Similarly, since F is the midpoint of BC, CF = (1/2)CB; since G is the midpoint of FC, CG = (1/2)CF = (1/4)CB. Triangles EGC and ABC share the same angle at C (∠ECG = ∠ACB, since E lies on CA and G lies on CB). The ratio of their areas is (CE/CA)·(CG/CB) = (1/4)·(1/4) = 1/16. So Area(ABC) : Area(EGC) = 16 : 1. Answer: E.
11
Infinite Geometric Series
medium
What does the value of the following expression get close to as n gets increasingly large? 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ⋯ + 1/2ⁿ
A
5/2
B
3/2
C
9/4
D
1.2
E
1
Show answer
Answer: E
This is an infinite geometric series with first term 1/2 and common ratio 1/2. The sum as n → ∞ is a/(1−r) = (1/2)/(1 − 1/2) = (1/2)/(1/2) = 1. Answer: E.
12
Repeating Decimals
medium
A, B, C, D are all positive integers. We know that A/B = 0.overline(24) and C/D = 0.overline(75). What is the simplified value of the following expression? A/B + C/D
A
2.5
B
2
C
1.2
D
0.99
E
1
Show answer
Answer: E
A repeating decimal 0.overline(ab) equals ab/99. So A/B = 24/99 = 8/33 and C/D = 75/99 = 25/33. Sum = 8/33 + 25/33 = 33/33 = 1. Answer: E.
13
Surface Area of a Cone
medium
In the figure below, there is a right cone with a base radius of CP = 3 meters and a height of CT = 4 meters. What is the whole surface area of the cone, including the base, in square meters?
A
10π m²
B
16π m²
C
24π m²
D
25π m²
E
28π m²
Show answer
Answer: C
The slant height l = √(r² + h²) = √(3² + 4²) = √25 = 5. Total surface area = πr² (base) + πrl (lateral surface) = π(3²) + π(3)(5) = 9π + 15π = 24π. Answer: C.
14
Volume / Rate Problems
hard
The cross-section and the shape of a public swimming pool are shown in the figure below, where the unit is meters. The pool is 50 meters long and 25 meters wide, with varied depths from 2 meters to 3 meters. The transition in between is smooth. If a pump can fill the pool at 25 cubic meters per hour, how long does it take to fill the pool to the brim?
A
50 hours
B
75 hours
C
100 hours
D
125 hours
E
150 hours
Show answer
Answer: D
The 50 m length splits into a 20 m section at constant depth 3 m, a 10 m transition section where depth changes smoothly (linearly) from 3 m to 2 m, and a 20 m section at constant depth 2 m. Over the linear transition, the average depth is (3+2)/2 = 2.5 m. Volume = width × [20×3 + 10×2.5 + 20×2] = 25 × [60 + 25 + 40] = 25 × 125 = 3125 m³. Time = Volume / rate = 3125 / 25 = 125 hours. Answer: D.
15
3D Geometry / Regular Tetrahedron
hard
Four spheres, which all have a radius of 1 centimeter, are arranged into a three-dimensional structure shown below. Each sphere touches all the others. What is the volume of the 3D shape, ABCD, formed by connecting the centers of the spheres?
A
(2√2)/3
B
(2√3)/5
C
(4√2)/5
D
(16√2)/3
E
√5/2
Show answer
Answer: A
Since each sphere has radius 1 and touches every other sphere, the distance between any two centers equals the sum of their radii, 1 + 1 = 2. So ABCD is a regular tetrahedron with edge length a = 2. The volume of a regular tetrahedron with edge a is V = a³/(6√2). For a = 2: V = 8/(6√2) = 4/(3√2) = (4√2)/6 = (2√2)/3. Answer: A.
Section B
5 Open-Ended Questions
(+5)
16
Ratio and Algebra
medium
Mark and Nancy both like collecting flower stickers. The ratio between Mark's collection and Nancy's collection is 6:9. If Nancy gives Mark 30 stickers, they would have the same number of stickers. How many stickers do they have in total?
Show answer
Answer: 300
The ratio 6:9 simplifies to 2:3, so let Mark have 2k and Nancy have 3k stickers. If Nancy gives Mark 30 stickers, Mark has 2k+30 and Nancy has 3k−30, and these are equal: 2k + 30 = 3k − 30 → k = 60. So Mark has 120 and Nancy has 180 stickers. Total = 120 + 180 = 300.
17
Probability
medium
There are a large number of marbles in a bag. 40% of the marbles are blue; the rest are all red. Mark is in need of a red marble, and he wants to make a random draw from the bag. If the marble is not red, he simply puts it back and continues. Let p represent the probability that it takes at least three random draws before he gets a red marble. What is 1000p?
Show answer
Answer: 160
Since a non-red marble is put back, each draw is an independent trial with P(red) = 0.6 and P(blue) = 0.4. 'At least three draws' means the first two draws are both not red (both blue). P = 0.4 × 0.4 = 0.16. So p = 0.16 and 1000p = 160.
18
Tangent Lines and Circles
medium
In the figure below, point A is the center of the circle. Points B and C are on the circle. BD is tangent to the circle at point B. AC is extended to intersect BD at point D. BD = 12 cm. The area of ΔABD is 30 cm². What is the length of line segment CD in cm?
Show answer
Answer: 8
Since BD is tangent to the circle at B, AB ⊥ BD. So triangle ABD is right-angled at B, and its area = (1/2)·AB·BD = 30. With BD = 12: (1/2)·AB·12 = 30 → 6·AB = 30 → AB = 5. Since AB is a radius, AC = AB = 5. By the Pythagorean theorem, AD = √(AB² + BD²) = √(5² + 12²) = √169 = 13. Since C lies on segment AD, CD = AD − AC = 13 − 5 = 8.
19
Area of Circles / Rings
medium
Three concentric circles are drawn, creating two rings, as shown in the figure. The radii of the circles are n, 3n and 9n inches, respectively, where n > 0. What is ratio between the area of the outer ring and the area of the inner ring?
Show answer
Answer: 9
The three circle areas are πn², π(3n)² = 9πn², and π(9n)² = 81πn². The inner ring (between the smallest and middle circle) has area 9πn² − πn² = 8πn². The outer ring (between the middle and largest circle) has area 81πn² − 9πn² = 72πn². The ratio of outer ring to inner ring is 72πn² : 8πn² = 9 : 1, so the ratio is 9.
20
Combinatorics
medium
A group of seven students are to stand in a row for a picture. In how many ways can the students be arranged such that the tallest and the shortest students in the group do not sit next to each other?
Show answer
Answer: 3600
Total arrangements of 7 students = 7! = 5040. Count the arrangements where the tallest and shortest ARE adjacent: treat them as a single block (which can be arranged internally in 2 ways), leaving 6 units to arrange in 6! ways, giving 2 × 6! = 2 × 720 = 1440. So the number of arrangements where they are NOT adjacent = 5040 − 1440 = 3600.
Section C
5 Open-Ended Questions
(+6)
21
Midpoints, Medians, and Similar Triangles
hard
In the following figure, points E and D are the midpoints of AB and AC, respectively. CE and BD intersect at point P. CG is parallel to AB, intersecting the extension of BD at G. If BP = 12 cm, what is the length of BG in cm?
Show answer
Answer: 36
Since D is the midpoint of AC and CG is parallel to AB, triangles ADB and CDG are congruent: AD = CD (D is a midpoint), ∠ADB = ∠CDG (vertical angles), and ∠DAB = ∠DCG (alternate interior angles since AB ∥ CG with AC as transversal). By ASA, ΔADB ≅ ΔCDG, so DB = DG, meaning D is also the midpoint of BG. Since E and D are midpoints of AB and AC, P is the centroid of triangle ABC where medians BD and CE meet, so P divides BD in ratio BP:PD = 2:1. With BP = 12, PD = 6, so BD = BP + PD = 18. Since D is the midpoint of BG, DG = BD = 18, and BG = BD + DG = 18 + 18 = 36.
22
Nested Fractions / Algebraic Simplification
hard
If A = [y/(y+1)] / (1 + [y/(y+1)]/(1 + y/(y+1))), B = [x/(x+2)] / (1 + [x/(x+2)]/(1 + x/(x+2))), C = [z/(z+3)] / (1 + [z/(z+3)]/(1 + z/(z+3))), what is the value of 1000(A + B + C) for y = 1, x = 2, and z = 3?
Show answer
Answer: 1125
Each expression has the same nested form: let s = u/(u+k). Then s/(1+s) = u/(2u+k), so 1 + s/(1+s) = (3u+k)/(2u+k), and the whole expression equals s divided by that: F = [u/(u+k)] · [(2u+k)/(3u+k)] = u(2u+k) / [(u+k)(3u+k)]. Here A uses u=y=1, k=1; B uses u=x=2, k=2; C uses u=z=3, k=3 — in every case u = k. Substituting u = k into the formula: F = u(2u+u)/[(u+u)(3u+u)] = u·3u/(2u·4u) = 3u²/8u² = 3/8, regardless of the actual value of u. So A = B = C = 3/8, and A+B+C = 9/8. Then 1000(A+B+C) = 1000 × 9/8 = 1125.
23
Systems of Equations
medium
In the figure below, three positive numbers, A, B and C are placed at the vertices of a triangle. The product between each pair of neighboring numbers is recorded on the corresponding side. What is √(A·B·C)?
Show answer
Answer: 70
From the figure, A·C = 196, B·C = 100, and A·B = 1225. Multiplying all three equations: (A·C)(B·C)(A·B) = A²B²C² = (ABC)². So ABC = √(196 × 100 × 1225) = √196 × √100 × √1225 = 14 × 10 × 35 = 4900. (Individually, A = 4900/100 = 49, B = 4900/196 = 25, C = 4900/1225 = 4, which checks: A·B=1225, B·C=100, A·C=196, all consistent.) Therefore √(A·B·C) = √4900 = 70.
24
Probability with Dice
medium
David has two six-sided fair dice, each of which has {1, 2, 3, 4, 5, 6} on its faces. David is interested in the product of the two numbers obtained when the two dice are rolled at random. Specifically, he wonders how likely he will get a square number as the product. He did 1000 trials. How many times should he expect to see a square number?
Show answer
Answer: 222
There are 36 equally likely outcomes. The perfect-square products (1, 4, 9, 16, 25, 36) arise from: product 1 → (1,1); product 4 → (1,4),(4,1),(2,2); product 9 → (3,3); product 16 → (4,4); product 25 → (5,5); product 36 → (6,6). That is 1+3+1+1+1+1 = 8 favorable outcomes out of 36, so P(square) = 8/36 = 2/9. Expected count over 1000 trials = 1000 × 2/9 ≈ 222.2, which rounds to 222.
25
Statistics: Capture-Recapture Estimation
medium
Greg has a large bag of soybeans. He is interested in estimating the number of soybeans using a statistical method he learned in class. He takes 200 soybeans from the bag and colored them red using a type of food coloring ink. Then, he puts the red beans back to the bag and mixes the beans thoroughly. Finally, he takes 50 beans, at random, from the bag, out of which he finds 4 red beans. How many soybeans should Greg expect to have in the bag?
Show answer
Answer: 2500
This is the capture-recapture (Lincoln–Petersen) method. The proportion of red beans in the random sample should approximate the proportion of red beans in the whole bag: 4/50 = 200/N, where N is the total number of soybeans. Solving: N = 200 × 50 / 4 = 10000/4 = 2500.