First Round Challenge: A 90 minute, 25 multiple choice question first round Challenge aimed at students year 13 or below. The problems on the Senior Maths Challenge are designed to make students think. Stimulating problems for both beginners and experienced problem-solvers.
Join our newsletter for updates and the Problem of The Day
When the expression (2² − 1) × (3² − 1) × (4² − 1) × (5² − 1) / (2 × 3) × (3 × 4) × (4 × 5) × (5 × 6) is simplified, which of the following is obtained?
The sequence of primes rounded to the nearest ten begins 0, 0, 10, 10, 10, 10, 20, 20. How many terms are equal to 40?
The base of a triangle is increased by 20% and its height is decreased by 15%. What happens to its area?
The figure shows a regular hexagon. How many parallelograms are there in the figure?
The diagram shows two congruent regular pentagons and a triangle. The angles marked G° are equal. What is the value of G?
In 2016, the world record for completing a 5000m three-legged race was 19 minutes and6 seconds. It was set by Damian Thacker and Luke Symonds in Sheffield.
What was their approximate average speed in km/h?
The diagram shows two symmetrically placed squares with sides of length 2 and 5. What is the ratio of the area of the small square to that of the shaded region?
The positive integer x is a solution of the equation (x ÷ 12) ÷ (15 ÷ x) = 20. What is the sum of the digits of x?
Three circles with radii 2, 3 and 3 touch each other, as shown in the diagram.
What is the area of the triangle formed by joining the centres of these circles?
The sum of four consecutive primes is itself prime. What is the largest of the four primes?
How many lines of three adjacent cells can be chosen from this grid, horizontally, vertically or diagonally, such that the sum oft he numbers in the three cells is a multiple of three?
Three points P, Q, and R are placed on the circumference of a circle with centre O. The arc lengths PQ, QR, and RP are in the ratio 1:2:3. In what ratio are the areas of the sectors OPQ, OQR, and ORP?
A sequence begins 2023, 2022, 1, . . . . After the first two terms, each term is the positive difference between the previous two terms. What is the value of the 25th term?
In 2021, a first-class postage stamp cost 85p and a second-class postage stamp cost 66p. To spend an exact number of pounds and buy at least one of each type, what is the smallest total number of stamps that should be purchased?
What is the value of 99 x (0.(49) − 0.(4))?
0.(49) = 0.49494949...
0.(4) = 0.4444444...
In the diagram, the outer hexagon is regular and has an area of 216. What is the shaded area?
When completed correctly, the cross number is filled with four three-digit numbers. What digit is * ?
A light-nanosecond is the distance that a photon can travel at the speed of light in one billionth of a second. The speed of light is 3 × 10⁸ ms⁻¹. How far is a light-nanosecond?
Alison has a set of ten fridge magnets showing the integers from 0 to 9 inclusive. In how many different ways can she split the set into five pairs so that the sum of each pair is a multiple of 5?
How many of the numbers 6, 7, 8, 9, 10 are factors of the sum 2^2024 + 2^2023 + 2^2022?
In a survey, people were asked to name their favourite fruit pie. The pie chart shows the outcome. What is the smallest number of people who could have been surveyed?
Wenlu, Xander, Yasser and Zoe make the following statements:
Wenlu: “Xander is lying.”
Xander: “ Yasser is lying.”
Yasser: “Zoe is telling the truth.”
Zoe: “Wenlu is telling the truth.”
What are the possible numbers of people telling the truth?
In the number triangle shown, each disc is to be filled with a positive integer. Each disc in the top or middle row contains the number which is the product of the two numbers immediately below. What is the value of n?
Alitta claims that if p is an odd prime then p² − 2 is also an odd prime. Which of the following values of p is a counterexample to this claim?
The greatest power of 7 which is a factor of 50! is 7^k (n! = 1×2×3×4×. . .× (n − 1) × n). What is k?
What is the sum of the digits of the integer which is equal to 6666666² - 3333333²?
For how many positive integers n is the remainder 6 when 111 is divided by n?
PQRST is a regular pentagon. The point U lies on ST such that ∠QPU is a right angle. What is the ratio of the interior angles in triangle PUT?
Three rugs have a combined area of 90 m². When they are laid down to cover completely a floor of area 60 m², the area which is covered by exactly two layers of rug is 12 m². What is the area of floor covered by exactly three layers of rug?
The points P (d, −d) and Q (12 − d, 2d − 6) both lie on the circumference of the same circle whose centre is the origin. What is the sum of the two possible values of d?
The diagram shows a square PQRS. A second square UVWX is drawn inside it, where U divides side PQ in the ratio 1:2. Similarly, a third square is drawn inside UVWX with its side divided in the same ratio. What fraction of the area of PQRS is shaded?
What is the smallest number of rectangles, each measuring 2 cm by 3 cm, which are needed to fit together without overlap to form a rectangle whose sides are in the ratio 5:4?
In Bethany’s class of 30 students, twice as many people played basketball as played football. Twice as many played football as played neither. Which of the following options could have been the number of people who played both?
The hare and the tortoise had a race over 100 m, in which both maintained constant speeds. When the hare reached the finish line, it was 75 m in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line did the hare and the tortoise meet?
Three dice, each showing numbers 1 to 6, are coloured red, blue and yellow respectively. Each of the dice is rolled once. The total of the numbers rolled is 10. In how many different ways can this happen?
G and H are midpoints of two adjacent edges of a cube. A trapezium-shaped cross-section is formed cutting through G, H and two further vertices, as shown. The edge-length of the cube is 2√2. What is the area of the trapezium?
An array of 25 equally spaced dots is drawn in a square grid as shown. Linda wants to draw a straight line through the diagram that passes through one other point.
The number M = 124563987 is the smallest number which uses all the non-zero digits once each and which has the property that none of the pairs of its consecutive digits makes a prime number. For example, the 5th and 6th digits of M make the number 63 which is not prime. N is the largest number which uses all the non-zero digits once each and which has the property that none of the pairs of its consecutive digits makes a prime number.
What are the 5th and 6th digits of N?
The shape shown is made by removing four equilateral triangles with side length 1 from a regular octagon with side length 1. What is the area of the shape?
A circle of radius r and a right-angled isosceles triangle are drawn such that one of the shorter sides of the triangle is a diameter of the circle. What is the shaded area?
How many solutions are there of the equation 1 + 2 sin X − 4 sin2 X − 8 sin3 X = 0 with 0° < X < 360°?
The numbers x and y are such that 3^x + 3^(y+1) = 5√3 and 3^(x+1) + 3^y = 3√3. What is the value of 3^x + 3^y?
The number 840 can be written as p!/q!, where p and q are positive integers less than 10. What is the value of p + q?
The expression (7n + 12) / (2n + 3) takes integer values for certain integer values of n. What is the sum of all such integer values of the expression?
How many pairs of real numbers (x, y) satisfy the simultaneous equations x² - y = 2022 and y² - x = 2022?
The diagram shows two overlapping triangles: Triangle ABC with interior angles 60°, 30°, and 90°, and Triangle DEF, which is a right-angled isosceles triangle. What is the ratio of the area of Triangle ABC to Triangle DEF?
Triangle LMN represents a right-angled field with LM = r, LX = p and XN = q. Jenny and Vicky walk at the same speed in opposite directions along the edge of the field, starting at X at the same time. Their first meeting is at M. Which of these expressions gives q in terms of p and r?
A square is inscribed inside a quadrant of a circle. The circle has a radius of 10. What is the area of the square?
Laura and Dina have a running race. Laura runs at constant speed and Dina runs n times as fast where n > 1. Laura starts B meters in front of Dina. What distance does Dina run before overtaking Laura?
Triangle PQR is equilateral. A semicircle with centre O is drawn with its diameter on PR so that one end is at P and the curved edge touches QR at X. The radius of the semicircle is √3. What is the length of QX?
The perimeter of a logo is created from two vertical straight edges, two small semicircles with horizontal diameters, and two large semicircles. Both the straight edges and the diameters of the small semicircles have a length of 2. The logo has rotational symmetry. What is the shaded area?
The numbers a and b satisfy the equations 2^a + 2^b = p and 2^a − 2^b = q. What is the value of 2^(a+b) in terms of p and q?
How many pairs of integers (x, y) satisfy the equation √x - √y + 23 = 2√2 - y?
A triangle with interior angles 60°, 45°, and 75° is inscribed in a circle of radius 2. What is the area of the triangle?
The length of a rectangular piece of paper is three times its width. The paper is folded so that one vertex lies on top of the opposite vertex, thus forming a pentagonal shape. What is the area of the pentagon as a fraction of the area of the original rectangle?
Three squares ABC, DEF, and GHI have vertices which sit on the sides of triangle JKL as shown. The squares have areas of 10, 90, and 40, respectively. What is the area of triangle JKL?
Let x be a real number. What is the minimum value of (x² - 4x + 3)(x² + 4x + 3)?
A square has its vertices on the edges of a regular hexagon. Two of the edges of the square are parallel to two edges of the hexagon, as shown in the diagram. The sides of the hexagon have length 1. What is the length of the sides of the square?
The numbers x, y, z, and w are all integers. x and y are variable, and z and w are constant and positive. The four integers are related by the equation xy = z(x + w). When y takes its maximum possible value, which expression is equal to y - x?
Saba, Rayan, and Derin are cooperating to complete a task. When all three work together, it takes 5 minutes to complete the task. How many minutes does it take for Derin to complete the task on his own?
What is the area of the part of the xy-plane within which x3y2 − x2y2 − xy4 + xy3 ≥ 0 and 0 ≤ x ≤ y?
A drinks carton is formed by arranging four congruent triangles as shown. PQ = QR = 4 cm and PR = PS = QR = PQ = 10 cm. What is the volume, in cm³, of the carton?
Five line segments of length 2, 2, 2, 1, and 3 connect two corners of a square as shown in the diagram. What is the shaded area?
Ask about our courses and offerings, and we will help you choose what works best for you.
