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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.
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?
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?
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?
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?
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?
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?
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?
The numbers x and y satisfy both of the equations.
23x + 977y = 2023 and 977x + 23y = 2977.
What is the value of x2 − y2?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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.
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?
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?
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?
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