Math

Instructions: Work in teams of three. Show full reasoning. Calculators are allowed only where stated. Time: 90 minutes.

Round 1: Core and Applied Challenges

1. A company sells notebooks at KES 120 each. If it reduces the price by 10%, it sells 25% more. What is the percentage change in total revenue?
2. A water tank is filled by two pipes. Pipe A can fill the tank in 4 hours, and Pipe B in 6 hours. How long will both take together to fill ¾ of the tank?
3. A rectangle and a triangle have equal areas. The rectangle measures 12 cm by 9 cm. The triangle’s base is 18 cm. Find its height.
4. A car accelerates from 20 m/s to 30 m/s in 5 seconds. Find its acceleration and distance covered during that time.
5. The nth term of a sequence is given by Tn = 2n² − 5n + 3. Find the first 5 terms and the sum of the first 10 terms.
6. A group of 3 friends buy 4 identical T-shirts and 2 pairs of jeans for KES 6,000. Another group buys 2 T-shirts and 3 pairs of jeans for KES 5,100. Find the cost of one T-shirt and one pair of jeans.
7. A circle with radius 7 cm is inscribed in a square. Find the area of the square outside the circle.
8. A trader mixes sugar costing KES 120/kg with sugar costing KES 180/kg to make 40 kg of mixture worth KES 150/kg. Find how much of each type he used.
9. A passenger train leaves Mombasa for Nairobi at 8:00 a.m. at 90 km/h. A goods train leaves an hour later at 120 km/h. At what time will the goods train overtake the passenger train?
10. The interior angle of a regular polygon is 165°. Find the number of sides.
11. The sum of three consecutive even numbers is 150. Find the numbers and their product.
12. A cylinder and a cone have equal bases and heights. If the radius is 5 cm and height 12 cm, find the difference in their volumes.
13. A number is increased by 25% and then decreased by 20%. Find the overall percentage change.
14. The area of a triangle is 48 cm². Its base is 12 cm and one side makes a 45° angle with the base. Find the length of the other side meeting at the 45° angle.
15. In a class, the ratio of boys to girls is 3:4. If 5 boys leave and 3 girls join, the ratio becomes 2:3. How many students were originally in the class?
16. Find the equation of a straight line passing through (2, 3) and perpendicular to 3x + 4y = 12.
17. A solid cube of side 8 cm is melted and recast into smaller cubes of side 2 cm. Find how many smaller cubes are formed.
18. The sum of an infinite geometric series is 16, and the first term is 10. Find the common ratio.
19. Find the smallest positive integer x that satisfies both: x ≡ 2 (mod 3) and x ≡ 3 (mod 5).
20. The diagonals of a rhombus are 16 cm and 12 cm. Find its perimeter.

Round 2: Advanced and Problem-Solving Challenges

21. A piece of wire 84 cm long is cut into two parts. One part is bent into a square and the other into a circle. Find the length of each part if the sum of their areas is minimum.
22. A function is defined by f(x) = x² − 5x + 6. Find the values of x for which f(x) = f⁻¹(x).
23. A box contains 5 red, 4 blue, and 3 green balls. Two balls are drawn one after another without replacement. Find the probability that both are red or both blue.
24. Two towns A and B are 150 km apart. A car leaves A at 8:00 a.m. towards B at 80 km/h. Another car leaves B at 9:00 a.m. towards A at 100 km/h. At what time do they meet?
25. Find the values of k for which 3x² + kx + 4 = 0 has equal roots.
26. The perimeter of a rectangle is 60 cm. If its length is 5 cm more than twice its width, find its area.
27. The line y = 2x + 3 touches a circle at (1, 5). Find the equation of the circle if its center lies on the x-axis.
28. A solid cone has radius 7 cm and height 24 cm. It is cut parallel to the base at half its height. Find the ratio of the volume of the smaller cone to the original cone.
29. Find all integer values of n for which (n + 1)(n − 3) < 0.
30. A machine depreciates at 20% per year. If its value after 3 years is KES 102,400, find its initial value.
31. A triangle has sides 8 cm, 15 cm, and 17 cm. Find the radius of the circle inscribed within the triangle.
32. A particle moves along a straight line such that its distance from a fixed point after t seconds is s = t³ − 6t² + 9t. Find its velocity when it changes direction.
33. Find the coordinates of the vertex and axis of symmetry for y = 2x² − 8x + 3.
34. The sum of n terms of an arithmetic sequence is given by Sₙ = n(3n + 1). Find its first term and common difference.
35. The mean of 9 numbers is 15. When one number is removed, the mean becomes 14. Find the number removed.
36. Three cubes have volumes in the ratio 1:8:27. Find the ratio of their total surface areas.
37. A bag contains 6 red, 5 blue, and 4 white marbles. Three are drawn at random. Find the probability that at least two are red.
38. The perimeter of an isosceles triangle is 36 cm. The base is 12 cm. Find the area of the triangle.
39. The quadratic 2x² − 3x − 2 = 0 has roots α and β. Find the value of α² + β².
40. A boat sails 15 km upstream and back in 3 hours. The river’s speed is 2 km/h. Find the speed of the boat in still water.