Download [PDF] 2021 UKMT UK IMC Questions Intermediate Mathematical Challenge United Kingdom Mathematics Trust

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Intermediate Mathematical Challenge 1 – 4 February 2021 Organised by the United Kingdom Mathematics Trust supported by England & Wales: Year 11 or below Scotland: S4 or below Northern Ireland: Year 12 or below 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; 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. Enquiries about the Intermediate Mathematical Challenge should be sent to: UK Mathematics Trust, School of Mathematics, University of Leeds, Leeds LS2 9JT T0113 365 1121 enquiry@ukmt.org.uk www.ukmt.org.uk

Intermediate Mathematical Challenge 1 – 4 February 20211. What is the value of 2021−2223 +2425 ? A 2122B 2223C 2324D 2425E 2526 2. The day before the day before yesterday was two days after the day before my birthday. Today is Thursday. On what day was my birthday? A SundayB MondayC TuesdayD WednesdayE Friday 3. What is the value of 2− (− 2− 2) − (− 2− (− 2− 2)) ? A 0B 2C 4D 6E 8 4. The diagram shows three squares, �� ��,� �� � and� �� � . Angles ���and�� � are 62 °and 74 °respectively. What is angle � � �? A 44 ° B 48° C 60° D 64° E 68° 5. April, May and June have 90 sweets between them. May has three-quarters of the number of sweets that June has. April has two-thirds of the number of sweets that May has. How many sweets does June have? A 60B 52C 48D 40E 36 6. Kai has begun to list, in ascending order, the positive integers which are notfactors of 240. What is the sixth number on Kai’s list? A 11B 13C 14D 15E 17 7. What is the value of (4 − 1 4 ) ÷ ( 2− 1 2 ) ? A 11 2 B 2C 21 2 D 3E 41 4 8.The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the centre � of one rectangle and the midpoint � of one edge of the other. What is the distance � �? A 12B 15C 18D 21E 24 9. How many of the following statements are true? A prime multiplied by a prime is always a prime. A square multiplied by a square is always a square. An odd number multiplied by an odd number is always an odd number. An even number multiplied by an even number is always an even number. A 0B 1C 2D 3E 4 © UK Mathematics Trust 2021 www.ukmt.org.uk � � � � � � � � � 62 ◦ 74◦ � � ������ ������

Intermediate Mathematical Challenge 1 – 4 February 202110. The prime factor decomposition of 2021 is 43×47 . What is the value of 53×57 ? A 2221B 2521C 2921D 3021E 3031 11. The line with equation �= 2� + 3is reflected in the �-axis. Which of the following is the equation of the new line? A �= 2� − 3 B�= −2� + 3 C�= 2� + 3 D�= 1 2 � + 3 E�= −2� − 3 12. Andrew calculates that 40%of50% of�is equal to 20%of30% of�, where �≠ 0. Which of the following is true? A �= 2 � 3 B �= 4 � 3 C �= 2� D�= 8 � 3 E �= 10 � 3 13. What is the remainder when 12 345×54 321 is divided by 9? A 0B 1C 2D 3E 4 14. The diagram shows a large square divided into squares of three different sizes. What percentage of the large square is shaded? A 61%B 59%C 57%D 55%E 53% 15.Patrick drives from P to Q at an average speed of 40 mph . His drive back from Q to P is at an average speed of 45 mphand takes two minutes less. How far, in miles, is it from P to Q? A 1.5B 6C 9D 12E 15 16. A semicircle is drawn on each side of a square, as shown. The square has sides of length 2� . What is the area of the resulting shape? A 2� 2 (� + 1) B�2 (� + 2) C2� 2 (2 � + 1) D�2 (� + 4) E 2� 2 (� + 2) 17. In the rectangle �� �� , the side �� is of length 2 and the side � � is of length 4. Points � and � lie inside the rectangle so that ���and� �� are equilateral triangles. What is the area of the quadrilateral � ���? A 6 − √ 3 2 B 8 3 C 4− 2√ 3 D4− √ 3 E 3 18. Which of these is closest in size to 1? A 0.? 9 ? 5 B1.? 0 ? 5 C0.? 96 ? 0 D1.? 04 ? 0 E0.9 ? 5 19. The diagram shows two overlapping rectangles, each measuring � by �. The area of overlap is exactly one-quarter of the total area of the figure. What is the ratio �:�? A 5 : 2 B4 : 1 C3 : 1 D2 : 1 E3 : 2 © UK Mathematics Trust 2021 www.ukmt.org.uk 2 ������ ������ ������

Intermediate Mathematical Challenge 1 – 4 February 202120. Two straight lines have equations �= �� +4and � �=�� −7, where �and �are constants. The two lines meet at the point (3, 1). What is the value of �? A 1B 2C 3D 4E 5 21.The diagram shows two congruent equilateral triangles whose overlap is a hexagon. The areas of the smaller triangles, which are also equilateral, are 1, 1, 9, 9, 16 and 16, as shown. What is the area of the inner hexagon? A 68B 58C 48D 38E 28 22. What is the result when we simplify the expression  1 + 1 �   1− 2 � + 1  1+ 2 � − 1 ? A 1 B 1 � (� + 1) C 1 ( � + 1) ( �− 1) D 1 � (� + 1) ( �− 1) E � + 1 � 23. The diagram shows a semicircle with centre � and radius 2 and a semicircular arc with diameter � �. Angle �� �is a right angle. What is the area of the shaded region? A �− 2 B 2C �D 3E 2� − 2 24. Sam writes on a white board the positive integers from 1 to 6 inclusive, once each. She then writes � additional fives and �sevens on the board. The mean of all the numbers on the board is then 5.3. What is the smallest possible value of �? A 7B 9C 11D 13E 15 25. Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it. How long, in seconds, would it take Thomas to walk down the broken escalator? A 30B 40C 45D 50E 60 © UK Mathematics Trust 2021 www.ukmt.org.uk 9 9 16 16 1 1 � � � � 2