maths Olympiad

1. A rectangular garden has a length that is 4 m more than its width. If the area is 96 m², what is the width (in meters)?

A. 6 B. 8 C. 10 D. 12

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Solution

Correct: B

Let width = x, then length = x + 4. Area = x(x + 4) = 96 → x² + 4x – 96 = 0. Factoring: (x + 12)(x – 8) = 0 ⇒ x = 8 (positive root).

2. A car travels 120 km in 2.5 hours and then 180 km in 3 hours. What is the average speed for the entire trip (in km/h)?

A. 50 B. 54 C. 60 D. 66

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Solution

Correct: B

Total distance = 120 + 180 = 300 km. Total time = 2.5 + 3 = 5.5 h. Average speed = 300 ÷ 5.5 ≈ 54.5 km/h → 54 km/h to nearest whole.

3. A cylindrical can must hold 1 liter (1000 cm³) of oil. If the radius is 5 cm, what is the minimum height (in cm) needed?

A. 10.0 B. 12.0 C. 12.7 D. 13.5

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Solution

Correct: C

Volume = πr²h → 1000 = π(5²)h → h = 1000/(25π) ≈ 12.7 cm.

4. A ladder 10 ft long leans against a wall. The foot of the ladder slides away at 1 ft/s. How fast is the top sliding down when the foot is 6 ft from the wall (in ft/s)?

A. 0.6 B. 0.75 C. 0.8 D. 1.0

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Solution

Correct: B

x² + y² = 10². Given dx/dt = 1, x = 6 → y = 8. Differentiate: 2x dx/dt + 2y dy/dt = 0 → dy/dt = –x/y = –6/8 = –0.75 ft/s (downward).

5. A water tank fills at 30 L/min and drains at 15 L/min. Starting empty, how many minutes until 450 L remain?

A. 15 B. 20 C. 25 D. 30

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Solution

Correct: D

Net rate = 30 – 15 = 15 L/min. Time = 450 ÷ 15 = 30 min.

6. The sum of two numbers is 50 and their product is 589. What is the smaller number?

A. 19 B. 23 C. 27 D. 31

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Solution

Correct: A

Let numbers be x and 50 – x. x(50 – x) = 589 → x² – 50x + 589 = 0. Quadratic formula gives x = 19 or 31 → smaller is 19.

7. A rectangle has perimeter 36 cm. If the length is 2 cm more than the width, what is the area (in cm²)?

A. 70 B. 72 C. 77 D. 80

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Solution

Correct: D

Let width = x, length = x + 2. 2(x + x + 2) = 36 → 4x + 4 = 36 → x = 8. Area = 8·10 = 80 cm².

8. A stone is thrown upward so that its height is h(t)=30t-5t² meters. When does it hit the ground (in seconds)?

A. 3 B. 4 C. 5 D. 6

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Solution

Correct: D

Ground means h = 0 → 30t – 5t² = 0 → 5t(6 – t) = 0 → t = 0 or 6 → t = 6 s.

9. A pizza shop charges $2 per topping plus a fixed price for a cheese pizza. If 3 toppings cost $11 total, how much is the cheese pizza alone (in dollars)?

A. 4 B. 5 C. 6 D. 7

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Solution

Correct: B

Let cheese price = C. C + 3·2 = 11 → C = 11 – 6 = 5 dollars.

10. A spherical balloon inflates so that its radius increases at 0.5 cm/s. When r = 6 cm, how fast is the volume increasing (in cm³/s)?

A. 36π B. 45π C. 54π D. 72π

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Solution

Correct: D

V = (4/3)πr³ → dV/dt = 4πr² dr/dt = 4π(6²)(0.5) = 72π cm³/s.

11. A train leaves station A at 60 km/h. Two hours later, another train leaves A at 80 km/h. How long (in hours) until the second train overtakes the first?

A. 4 B. 5 C. 6 D. 8

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Solution

Correct: C

Let t = time for faster train. Distance equal: 80t = 60(t + 2) → 80t = 60t + 120 → 20t = 120 → t = 6 h.

12. A square and a circle have the same perimeter. If the square’s side is 11 cm, what is the radius of the circle (in cm, to one decimal)?

A. 6.5 B. 7.0 C. 7.5 D. 8.0

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Solution

Correct: B

Perimeter square = 4·11 = 44 cm → circumference circle = 44 = 2πr → r = 44/(2π) ≈ 7.0 cm.

13. A company sells x items at price (100 – x) dollars each. Which x maximizes revenue?

A. 40 B. 50 C. 60 D. 70

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Solution

Correct: B

Revenue R = x(100 – x) = –x² + 100x. Vertex of parabola at x = –b/(2a) = 100/2 = 50.

14. A tap fills a 600 L tank in 20 min. A second tap fills it in 30 min. How many minutes if both work together?

A. 10 B. 12 C. 15 D. 18

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Solution

Correct: B

Rates: 600/20 = 30 L/min and 600/30 = 20 L/min. Combined 50 L/min → time = 600/50 = 12 min.

15. A curve has slope dy/dx = 3x² – 4x. If it passes through (1, 2), what is y when x = 2?

A. 2 B. 3 C. 4 D. 5

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Solution

Correct: B

Integrate: y = x³ – 2x² + C. At x = 1, y = 2 → 1 – 2 + C = 2 → C = 3. At x = 2: y = 8 – 8 + 3 = 3.

16. A 15 m wire is bent into a rectangle with length 3 m more than width. What is the width (in meters)?

A. 2.25 B. 2.50 C. 2.75 D. 3.00

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Solution

Correct: A

Let width = x, length = x + 3. Perimeter 2(x + x + 3) = 15 → 4x + 6 = 15 → 4x = 9 → x = 2.25 m.

17. A bacteria culture doubles every 3 hours. Starting from 200 cells, how many hours to reach 6400 cells?

A. 9 B. 12 C. 15 D. 18

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Solution

Correct: C

Growth factor 6400/200 = 32 = 2⁵ → 5 doubling periods. 5·3 = 15 h.

18. A triangle has base 10 cm and height decreasing at 2 cm/min. How fast is the area decreasing when h = 6 cm (in cm²/min)?

A. 8 B. 10 C. 12 D. 14

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Solution

Correct: B

A = (1/2)bh → dA/dt = (1/2)b dh/dt = (1/2)(10)(–2) = –10 cm²/min → decreasing at 10 cm²/min.

19. A store offers 20% off then an extra 10% off the reduced price. What percent of the original price is the final price?

A. 68 B. 70 C. 72 D. 75

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Solution

Correct: C

Multiply factors: 0.8·0.9 = 0.72 → 72% of original.

20. A particle moves along the x-axis with velocity v(t)=6t-3t² m/s. At t = 0 it is at x = 0. Where is it at t = 2 s (in meters)?

A. 2 B. 4 C. 6 D. 8

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Solution

Correct: B

x(t)=∫v dt=3t²-t³+C. C=0. At t=2: x=3·4-8=12-8=4 m.