UKMT

Grey Kangaroo

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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Boundaries
Instructions

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.

Problems

Which of the shapes below cannot be divided into two trapeziums by a single straight  line?

2023

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?

2022

What is the sum of the largest three-digit multiple of 4 and the smallest four-digit multiple of 3?

2023

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?

2022

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?

2023

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?

2022

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?

2023

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?

2022

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?

2023

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?

2022

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?

2023

How many positive integers between 100 and 300 have only odd digits?

2022

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?

2023

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?

2022

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?

2023

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?

2022

The diagram shows five equal semicircles and the lengths of some line segments. What is the radius of the semicircles?

2023

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?

2022

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?

2023

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?

2022

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?

2023

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?

2022

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?

2023

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?

2022

In the diagram shown, sides PQ and PR are equal. Also ∠QPR = 40◦ and ∠TQP = ∠SRQ. What is the size of ∠TUR?

2023

The area of the intersection of a triangle and a circle is 45% of the total area of the diagram. The area of the triangle outside the circle is 40% of the total area of the diagram. What percentage of the circle lies outside the triangle?

2022

Tom, John and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points, as shown. How many points did Lily score?

2023

Jenny decided to enter numbers into the cells of a 3 × 3 table so that the sum of the numbers in all four possible 2 × 2 cells will be the same. The numbers in three of the corner cells have already been written, as shown. Which number should she write in the fourth corner cell?

2022

The diagram shows a smaller rectangle made from three squares, each of area 25 cm2 , inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in cm2 , of the larger rectangle?

2023

The villages P, Q, R and S are situated, not necessarily in that order, on a long straight road. The distance from P to R is 75 km, the distance from Q to S is 45 km and the distance from Q to R is 20 km. Which of the following could not be the distance, in km, from P to S?

2022

The sum of 2023 consecutive integers is 2023. What is the sum of digits of the largest of these integers?

2023

The large rectangle W XY Z is divided into seven identical rectangles, as shown. What is the ratio W X : XY?

2022

Some beavers and some kangaroos are standing in a circle. There are three beavers in total and no beaver is standing next to another beaver. Exactly three kangaroos stand next to another kangaroo. What is the number of kangaroos in the circle?

2023

You can choose four positive integers X, Y, Z and W. What is the maximum number of odd numbers you can obtain from the six sums X + Y, X + Z, X + W, Y + Z, Y + W and Z + W?

2022

Snow White organised a chess competition for the seven dwarves, in which each dwarf played one game with every other dwarf. On Monday, Grumpy played 1 game, Sneezy played 2, Sleepy 3, Bashful 4, Happy 5 and Doc played 6 games. How many games did Dopey play on Monday?

2023

Marc always cycles at the same speed and he always walks at the same speed. He can cover the round trip from his home to school and back again in 20 minutes when he cycles and in 60 minutes when he walks. Yesterday Marc started cycling to school but stopped and left his bike at Eva’s house on the way before finishing his journey on foot. On the way back, he walked to Eva’s house, collected his bike and then cycled the rest of the way home. His total travel time was 52 minutes. What fraction of his journey did Marc make by bike?

2022

Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this?

2023

A builder has two identical bricks. She places them side by side in three different ways, as shown. The surface areas of the three shapes obtained are 72, 96 and 102. What is the surface area of the original brick?

2022

There were twice as many children as adults sitting round a table. The age of each person at the table was a positive integer greater than 1. The sum of the ages of the adults was 156. The mean age of the children was 80% less than the mean age of the whole group. What the sum of the ages of the children?

2023

Carl wrote a list of 10 distinct positive integers on a board. Each integer in the list, apart from the first, is a multiple of the previous integer. The last of the 10 integers is between 600 and 1000. What is this last integer?

2022

Martin is standing in a queue. The number of people in the queue is a multiple of 3. He notices that he has as many people in front of him as behind him. He sees two friends, both standing behind him in the queue, one in 19th place and the other in 28th place. In which position in the queue is Martin?

2023

What is the smallest number of cells that need to be coloured in a 5 × 5 square grid so that every 1 × 4 or 4 × 1 rectangle in the grid has at least one coloured cell?

2022

Some mice live in three neighbouring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram ?

2023

Mowgli asked a snake and a tiger what day it was. The snake always lies on Monday, Tuesday and Wednesday but tells the truth otherwise. The tiger always lies on Thursday, Friday and Saturday but tells the truth otherwise. The snake said “Yesterday was one of my lying days”. The tiger also said “Yesterday was one of my lying days”. What day of the week was it?

2022

Bart wrote the number 1015 as a sum of numbers using only the digit 7. He used a 7 a total of 10 times, including using the number 77 three times, as shown. Now he wants to write the number 2023 as a sum of numbers using only the digit 7, using a 7 a total of 19 times. How many times will the number 77 occur in the sum?

2023

Several points were marked on a line. Renard then marked another point between each pair of adjacent points on the line. He performed this process a total of four times. There were then 225 points marked on the line. How many points were marked on the line initially?

2022

Jake wrote six consecutive numbers on six white pieces of paper, one number on each piece. He stuck these bits of paper onto the top and bottom of three coins. Then he tossed these three coins three times. On the first toss, he saw the numbers 6, 7 and 8 and then coloured them red. On the second toss, the sum of the numbers he saw was 23 and on the third toss the sum was 17. What was the sum of the numbers on the remaining three white pieces of paper?

2023

An isosceles triangle PQR, in which PQ = PR, is split into three separate isosceles triangles, as shown, so that PS = SQ, RT = RS and QT = RT. What is the size, in degrees, of angle QPR?

2022

A rugby team scored 24 points, 17 points and 25 points in the seventh, eighth and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th game for their mean number of points-per-game to exceed 22?

2023

There are 2022 kangaroos and some koalas living across seven parks. In each park, the number of kangaroos is equal to the total number of koalas in all the other parks. How many koalas live in the seven parks in total?

2022

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