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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.
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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?
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?
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.
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?
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?
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?
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?
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?
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?
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?
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?
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?
For how many positive integers n is the remainder 6 when 111 is divided by n?
What is the sum of the digits of the integer which is equal to 6666666² - 3333333²?
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?
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?
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?
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 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?
How many of the numbers 6, 7, 8, 9, 10 are factors of the sum 2^2024 + 2^2023 + 2^2022?
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?
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?
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