A 60 minute, 25 multiple choice Challenge.
It encourages mathematical reasoning, precision of thought and fluency to make students think.
The problems on the Intermediate Maths Challenge are designed to make students think, most are accessible yet still challenge those with more experience.
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Bronze: ≥47
Silver: ≥61
Gold ≥77
Follow on Olympiads:
Year 9 and Below: Grey Kangaroo ≥72
Year 9 and Below: Cayley Olympiad ≥101
Year 10 and 11: Pink Kangaroo ≥84
Year 10: Hamilton Olympiad ≥106
Year 11: Maclaurin Olympiad ≥110
1. Do not open the paper until the invigilator tells you to do so.
2. Time allowed: 60 minutes. No answers, or personal details, may be entered after the allowed time is over.
3. The use of blank or lined paper for rough working is allowed; squared paper, calculators and measuring instruments are forbidden.
4. Use a B or an HB non-propelling pencil. Mark at most one of the options A, B, C, D, E on the Answer Sheet for each question. Do not mark more than one option.
5. Do not expect to finish the whole paper in the time allowed. The questions in this paper have been arranged in approximate order of difficulty with the harder questions towards the end. You are not expected to complete all the questions during the time. You should bear this in mind when deciding which questions to tackle.
6. Scoring rules:
5 marks are awarded for each correct answer to Questions 1-15;
6 marks are awarded for each correct answer to Questions 16-25;
Each incorrect answer to Questions 16-20 loses 1 mark;
Each incorrect answer to Questions 21-25 loses 2 marks.
7. Your Answer Sheet will be read by a machine. Do not write or doodle on the sheet except to mark your chosen options. The machine will read all black pencil markings even if they are in the wrong places. If you mark the sheet in the wrong place, or leave bits of eraser stuck to the page, the machine will interpret the mark in its own way.
8. The questions on this paper are designed to challenge you to think, not to guess. You will gain more marks, and more satisfaction, by doing one question carefully than by guessing lots of answers. This paper is about solving interesting problems, not about lucky guessing.
What is the difference between the smallest two-digit prime and the largest two-digit prime?
The diagram shows the square PQRS and T, the midpoint of the side PQ.
What fraction of the area of the square PQRS is shaded?
The diagram shows a square, its two diagonals and two line segments, each of which connects two midpoints of sides of the square.
What fraction of the area of the square is shaded?
The shorter sides of a right-angled triangle have lengths √5 and √12.
What is the length of the hypotenuse?
We know that 1 + 2 + 3 + 4 = 10.
It is also true that 13 + 23 + 33 + 43 = 10n for some integer n.
What is this integer?
The ages of Grannie’s seven grandchildren are consecutive positive integers.
The youngest three grandchildren have a mean age of 6.
What is the mean age of the oldest three grandchildren?
To draw a 3 by 3 square grid you need 8 straight lines, as shown.
How many straight lines do you need to draw a n by n square grid?
Four of these points lie on a circle.
Which of the points does not lie on that circle?
The ‘Penny’s Puddings’ company uses one tonne of rice to make twenty-five thousand cans of rice pudding. Each tonne of rice contains approximately fifty million grains ofrice. Approximately how many grains of rice are there in a can of Penny’s rice pudding?
Merryn wrote down the numbers 2, 0, 2, 3 and one further number.
What was the median of her five numbers?
Three sectors of a circle are removed from a regular hexagon to form the shaded shape shown.
Each sector has perimeter 18 mm. What is the perimeter, in mm, of the shaded shape formed?
Eight of the digits from 0 to 9 inclusive are used to fill the cells of the crossnumber. What is the sum of the two digits which are not used?
Jill was given a large jar of jam. She gave one sixth of the jam to Jan. Jill then gave one thirteenth of the remaining jam to Jas. Jill was left with 1 kg of jam.
What was the weight, in kg, of the jam in Jill’s jar at the start?
A picture, together with its frame, makes a square of side length 80 cm. The frame is 4 cm wide.
What percentage of the area of the square is covered by the frame?
In the diagram, PQRS is a square, PST is an equilateral triangle and SRUVW is a regular pentagon.
What is the size of angle WTS?
The mean of p and q is 13; the mean of q and r is 16; the mean of r and p is 7.
What is the mean of p, q and r?
When I increase a certain number by 20%, I get twice as much as when I decrease 20 less than the number by 20%.
What is that number?
A regular octagon PQRSTUV has sides of length 2 cm.
When I shade the rectangles PQTU and RSVW, four small triangles inside the octagon remain unshaded. What is the total area, in cm2, of these four triangles?
Going clockwise around a quadrilateral, its interior angles are in the ratio 6 : 7 : 8 : 9.
Which of the following is a true statement about the quadrilateral?
How many of the following polygons could exist?
A triangle with all three sides the same length, but three different interior angles.
A quadrilateral with all four sides the same length, but four different interior angles.
A pentagon with all five sides the same length, but five different interior angles.
Carrie the cat and Barrie the bat together weigh 4000 g more than Rollie the rat.
Barrie and Rollie together weigh 2000 g less than Carrie.
Carrie and Rollie together weigh 3000 g more than Barrie.
What is the weight, in grams, of Rollie the rat?
The sum of the lengths of the three sides of a right-angled triangle is 16 cm.
The sum of the squares of the lengths of the three sides of the triangle is 98 cm2.
What is the area, in cm2, of the triangle?
Factorial n, written n!, is defined by: n! = 1 × 2 × 3 × · · · × n.
What is the remainder when 1!+2!+3!+4!+5!+6!+7!+8!+9!+10! is divided by 5?
A 3 by 2 rectangle is split into four congruent right-angled triangles, as shown in the left-hand diagram.
Those four triangles are rearranged to form a rhombus, as shown in the right-hand diagram.
What is the ratio of the perimeter of the rectangle to the perimeter of the rhombus?
The point P (−1, 4) is reflected in the y-axis to become Q. The point Q is reflected in the line y = x to become R.
The point R is reflected in the x-axis to become S. What is the area of quadrilateral PQRS?
What is the positive difference between the numerator and the denominator when the expression shown is written as a single fraction in its simplest form?
In the grid shown the three non-zero numbers in each row, each column and each diagonal multiply to give the same product.
What is the value of x?
I roll two standard six-sided fair dice. At least one of the scores obtained on the dice is 3. What is the probability that both of the scores on the dice are 3?
A shop sign says, “T-shirts. Three for the price of two. Equivalent to a saving of £5.50 on each T-shirt.”
Using this special offer, what is the cost of three T-shirts?
A semicircle of radius 3 units is drawn on one edge of a right-angled triangle, and a semicircle of radius 4 units is drawn on another edge.
The semicircles intersect on the hypotenuse of the triangle, as shown.What is the shaded area, in square units?
The diagram shows a square of side 4 cm with four identical semi-circles drawn with their centres at the mid-points of the sides.
The four semi-circles each touch two other semi-circles, as shown.
What is the shaded area, in cm2?
The numbers x and y satisfy both of the equations.
23x + 977y = 2023 and 977x + 23y = 2977.
What is the value of x2 − y2?
It is possible to choose, in two different ways, six different integers from 1 to 9 inclusive such that their product is a square.
Let the two squares so obtained be p2and q2, where p and q are both positive.
What is the value of p + q?
When a cube is cut into two pieces with a single plane cut, two polyhedra are obtained. Which of these polyhedra cannot be obtained in this way?
A circle is inscribed in a semicircle with centre O and diameter AB.
The centre of the circle is the point P, wherePA = PO.
What is the ratio of the radius of the circle to the radius of the semicircle?
A rectangle PQRS has side-lengths a and b, with a < b. The rectangle PTUV has side-lengths c and d, with c < d. Also, a < d and c < b, as shown. The sides RS and TU cross at X.
Which of these conditions guarantees that Q, X and V lie on a straight line?
The diagram shows a regular hexagon RSTUVW.
The area of the shaded pentagon RSTPQ is one quarter of the area of the hexagon. Jay and Kay walk around the hexagon from P to Q, Jay going clockwise and Kay anticlockwise.
What is the ratio of the distance Jay walks to the distance Kay walks?
The diagram shows two unshaded circles which touch each other and also touch a larger circle.
Chord PQ of the larger circle is a tangent to both unshaded circles. The length of PQ is 6 units.
What is the area, in square units, of the shaded region?
A gold coin is worth x% more than a silver coin. The silver coin is worth y% less than the gold coin.
Both x and y are positive integers.How many possible values for x are there?
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