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As follow-on Rounds to the Intermediate Maths Challenge, the Grey and Pink Kangaroos are 60 minute, 25 multiple choice challenges. Entry to the Grey and Pink Kangaroos is by invitation based on a qualfying IMC score, or by discretionary entry. Several thousand UK-based students qualify from the IMC each year. Open to UK schools only.
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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;
In this paper you will not lose marks for getting answers wrong.
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, or reject the answer sheet.
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.
In the diagram shown, sides PQ and PR are equal. Also ∠QPR = 40◦ and ∠TQP = ∠SRQ. What is the size of ∠TUR?
The numbers 1 to 8 are to be placed, one per circle, in the circles shown. The number next to each arrow shows what the product of the numbers in the circles on that straight line should be. What will be the sum of the numbers in the three circles at the bottom of the diagram?
A square with side-length 10 cm long is drawn on a piece of paper. How many points on the paper are exactly 10 cm away from two of the vertices of this square?
Werner wrote a list of numbers with sum 22 on a piece of paper. Ria then subtracted each of Werner’s numbers from 7 and wrote down her answers. The sum of Ria’s numbers was 34. How many numbers did Werner write down?
Matchsticks can be used to write digits, as shown in the diagram. How many different positive integers can be written using exactly six matchsticks in this way?
In my office there are two digital 24-hour clocks. One clock gains one minute every hour and the other loses two minutes every hour. Yesterday I set both of them to the same time but when I looked at them today, I saw that the time shown on one was 11:00 and the time on the other was 12:00. What time was it when I set the two clocks?
Some edges of a cube are to be coloured red so that every face of the cube has at least one red edge. What is the smallest possible number of edges that could be coloured red?
Three sisters, whose average age is 10, all have different ages. The average age of one pair of the sisters is 11, while the average age of a different pair is 12. What is the age of the eldest sister?
The diagram shows five equal semicircles and the lengths of some line segments. What is the radius of the semicircles?
Tony the gardener planted tulips and daisies in a square flowerbed of side-length 12 m, arranged as shown. What is the total area, in m2 , of the regions in which he planted daisies?
Theodorika wrote down four consecutive positive integers in order. She used symbols instead of digits. She wrote the first three integers as □ ♢ ♢, ♥ △ △, ♥ △ □. What would she write in place of the next integer in the sequence?
On a standard dice, the sum of the numbers of pips on opposite faces is always 7. Four standard dice are glued together as shown. What is the minimum number of pips that could lie on the whole surface?
Evita wants to write the numbers 1 to 8 in the boxes of the grid shown, so that the sums of the numbers in the boxes in each row are equal and the sums of the numbers in the boxes in each column are equal. She has already written numbers 3, 4 and 8, as shown. What number should she write in the shaded box?
Anna has five circular discs, each of a different size. She decides to build a tower using three of her discs so that each disc in her tower is smaller than the disc below it. How many different towers could Anna construct?
There are five big trees and three paths in a park. It has been decided to plant a sixth tree so that there are the same number of trees on either side of each path. In which region of the park should the sixth tree be planted?
John has 150 coins. When he throws them on the table, 40% of them show heads and 60% of them show tails. How many coins showing tails does he need to turn over to have the same number showing heads as showing tails?
In the equation on the right there are five empty squares. Sanja wants to fill four of them with plus signs and one with a minus sign so that the equation is correct. Where should she place the minus sign?
Kristina has a piece of transparent paper with some lines marked on it. She folds it along the central dashed line, as indicated. What can she now see?
The front number plate of Max’s car fell off. He put it back upside down but luckily this didn’t make any difference. Which of the following could be Max’s number plate?
Werner wants to write a number at each vertex and on each edge of the rhombus shown. He wants the sum of the numbers at the two vertices at the ends of each edge to be equal to the number written on that edge. What number should he write on the edge marked with the question mark?
Kanga likes jumping on the number line. She always makes two large jumps of length 3, followed by three small jumps of length 1, as shown, and then repeats this over and over again. She starts jumping at 0. Which of these numbers will Kanga land on?
What is the sum of the largest three-digit multiple of 4 and the smallest four-digit multiple of 3?
Beate rearranges the five numbered pieces shown to display the smallest possible nine-digit number. Which piece does she place at the right-hand end?
Which of the shapes below cannot be divided into two trapeziums by a single straight line?
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