MATHEMATICAL CHALLENGE

Transcription

1 The Scottish Mathematical Council MATHEMATICAL CHALLENGE Entries must be the unaided efforts of individual pupils. Solutions must include explanations and answers without explanation will be given no credit. Do not feel that you must hand in answers to all the questions. CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE ARE The Edinburgh Mathematical Society, Professor L E Fraenkel, The London Mathematical Society and The Scottish International Education Trust. The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and other assistance from schools, universities and education authorities. Particular thanks are due to the Universities of Aberdeen, Edinburgh, Glasgow, St Andrews, Strathclyde, and to Preston Lodge High School, Bearsden Academy, Beaconhurst School, St Aloysius College and Turriff Academy. Junior Division: Problems 2 J1. You are given three rods of lengths 1, 3 9 units 1 unit and 9 units. Using these rods, you could measure 7 units as shown. Show how you could measure each 3 units whole number length up to 13 units. 7 units By adding a fourth rod, it is possible to measure all whole number lengths up to 40 units. What is the length of this extra rod? Explain your answer. J2. You have three boxes, each containing two identically wrapped Easter eggs. One box contains two milk chocolate eggs (M), one contains two plain chocolate eggs (P) and the third contains one milk chocolate egg and one plain chocolate egg. The boxes are labelled MM, PP or MP according to their contents. However, someone has switched all the labels so that every box is now incorrectly labelled. You are allowed to take out one egg at a time from any box, check what type it is and put it back. By doing this you can correctly label all three boxes. What is the smallest number of eggs you would need to check in order to label the boxes correctly? Explain your answer. J3. Take all the prime numbers between 30 and 60 and place them in a row in such a way that: (a) the sum of the two largest numbers and the numbers between them in the row is 233; (b) the sum of the smallest prime, the middle one in size and the numbers in between them in the row is 133; (c) the difference between the first and last primes in the row is 6; (d) the difference between the second and sixth primes in the row is also 6. Justify your conclusion. SEE OVER FOR QUESTIONS J4 and J5.

2 SMC SURNAME OTHER NAME(S) (underline the one you prefer) SCHOOL Mathematical Challenge Problems 2 JUNIOR DIVISION PLEASE USE CAPITALS TO COMPLETE FOR OFFICIAL USE Marker Marks AGE YEAR OF STUDY S Total C U T A L O N G H E R E Please write your solutions on A4 paper and staple the above form to them. PLEASE WRITE YOUR NAME ON EVERY PAGE. Send your entry directly or through your school to : Simon Malham School of Mathematical and Computer Science Heriot Watt University Edinburgh EH14 4AS For further information on the competition, please see the Information Circular, which has been distributed to all secondary schools. Please contact the local organiser, whose name and address are given above, if you require a further copy. J4. Four cards, each numbered with a different whole number, are placed face down. Four people, Gavin, Jack, Katie and Luke, in turn select two of these cards, write down their total, and then replace the two cards. Gavin's total is 6, Jack's 9, Katie's 12 and Luke's 15. Two of the cards are then turned over and their total is 11. Determine the numbers on each of the cards. J5. An old-fashioned rectangular billiard table has only four pockets, one at each corner. The lengths of the sides of the table form a whole number ratio. Show that, if the ratio is 5 : 2 and a ball is hit from one corner at an angle of 45, it will land in a pocket after 5 rebounds. If the ratio of the sides were m : n, where m and n are different whole numbers, with no common factor, and the ball were hit from a corner at an angle of 45, show that the ball would always drop into a pocket after a number of rebounds. How many rebounds would there be in this case? CLOSING DATE FOR RECEIPT OF SOLUTIONS : END OF PROBLEM SET 2 18th February 2011 Look on the SMC web site: for information about Mathematical Challenge

3 The Scottish Mathematical Council MATHEMATICAL CHALLENGE Entries must be the unaided efforts of individual pupils. Solutions must include explanations and answers without explanation will be given no credit. Do not feel that you must hand in answers to all the questions. CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE ARE The Edinburgh Mathematical Society, Professor L E Fraenkel, The London Mathematical Society and The Scottish International Education Trust. The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and other assistance from schools, universities and education authorities. Particular thanks are due to the Universities of Aberdeen, Edinburgh, Glasgow, St Andrews, Strathclyde, and to Preston Lodge High School, Bearsden Academy, Beaconhurst School, St Aloysius College and Turriff Academy. Middle Division: Problems 2 M1. Four cards, each numbered with a different whole number, are placed face down. Four people, Gavin, Jack, Katie and Luke, in turn select two of these cards, write down their total, and then replace the two cards. Gavin's total is 6, Jack's 9, Katie's 12 and Luke's 15. Two of the cards are then turned over and their total is 11. Determine the numbers on each of the cards. M2. An old-fashioned rectangular billiard table has only four pockets, one at each corner. The lengths of the sides of the table form a whole number ratio. Show that, if the ratio is 5 : 2 and a ball is hit from one corner at an angle of 45, it will land in a pocket after 5 rebounds. If the ratio of the sides were m : n, where m and n are different whole numbers, with no common factor, and the ball were hit from a corner at an angle of 45, show that the ball would always drop into a pocket after a number of rebounds. How many rebounds would there be in this case? M3. A farmer was having cash-flow problems and was discussing his options with his wife. If we sell 75 chickens we will bring in some money and my existing stock of feed will last an extra 20 days. But if we buy an additional 100 chickens, we will get money from the extra eggs, but my existing stock of feed will last 15 days less. Exactly how many chickens do you currently have? asked his wife. What is the answer to his wife s question and why is this the answer? M4. The shape of a fifty-pence piece is based on a regular heptagon which is a 7-sided polygon. The distance between each vertex and each of its two nearly opposite vertices is 1 unit. The perimeter of the coin is formed by circular arcs of radius 1 unit which are centred on each vertex, and join the two nearly opposite vertices. Find the length of the perimeter of the coin. SEE OVER FOR QUESTION M5.

4 SMC SURNAME OTHER NAME(S) (underline the one you prefer) SCHOOL Mathematical Challenge Problems 2 MIDDLE DIVISION PLEASE USE CAPITALS TO COMPLETE FOR OFFICIAL USE Marker Marks AGE YEAR OF STUDY S Total C U T A L O N G H E R E Please write your solutions on A4 paper and staple the above form to them. PLEASE WRITE YOUR NAME ON EVERY PAGE. Send your entry directly or through your school to : Simon Malham School of Mathematical and Computer Science Heriot Watt University Edinburgh EH14 4AS For further information on the competition, please see the Information Circular, which has been distributed to all secondary schools. Please contact the local organiser, whose name and address are given above, if you require a further copy. M5. A rabbit's burrow is at A and he knows that there are carrots in a garden at B, across a road, which is 10m wide. The burrow is 20m from the nearer edge of the road and the carrots are 30m beyond the other edge as shown in the diagram. The straight line distance from A to B is 80m. B 30m 20m A 10m The rabbit is wary of crossing the road and knows from past experience that he must cross directly across the road, not askew. What is the length of the shortest possible route for the rabbit from the burrow to the carrots? CLOSING DATE FOR RECEIPT OF SOLUTIONS : END OF PROBLEM SET 2 18th February 2011 Look on the SMC web site: for information about Mathematical Challenge

5 The Scottish Mathematical Council MATHEMATICAL CHALLENGE Entries must be the unaided efforts of individual pupils. Solutions must include explanations and answers without explanation will be given no credit. Do not feel that you must hand in answers to all the questions. CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE ARE The Edinburgh Mathematical Society, Professor L E Fraenkel, The London Mathematical Society and The Scottish International Education Trust. The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and other assistance from schools, universities and education authorities. Particular thanks are due to the Universities of Aberdeen, Edinburgh, Glasgow, St Andrews, Strathclyde, and to Preston Lodge High School, Bearsden Academy, Beaconhurst School, St Aloysius College and Turriff Academy. Senior Division: Problems 2 S1. The shape of a fifty-pence piece is based on a regular heptagon which is a 7- sided polygon. The distance between each vertex and each of its two nearly opposite vertices is 1 unit. The perimeter of the coin is formed by circular arcs of radius 1 unit which are centred on each vertex, and join the two nearly opposite vertices. Find the length of the perimeter of the coin. S2. A rabbit's burrow is at A and he knows that there are carrots in a garden at B, across a road, which is 10m wide. The burrow is 20m from the nearer edge of the road and the carrots are 30m beyond the other edge as shown in the diagram. The straight line distance from A to B is 80m. B 30m 20m A 10m The rabbit is wary of crossing the road and knows from past experience that he must cross directly across the road, not askew. What is the length of the shortest possible route for the rabbit from the burrow to the carrots? S3. One disc of 20 cm diameter and one of 10 cm diameter are cut from a disc of plywood of diameter 30 cm. What is the diameter of the largest disc that can be cut from the wood that remains? (Ignore the thickness of the saw cut.) S4. Calculate Find the value of the square of the number consisting of one million sixes, followed by one seven. Justify your answer. SEE OVER FOR QUESTION S5.

6 SMC SURNAME OTHER NAME(S) (underline the one you prefer) SCHOOL Mathematical Challenge Problems 2 SENIOR DIVISION PLEASE USE CAPITALS TO COMPLETE FOR OFFICIAL USE Marker Marks AGE YEAR OF STUDY S Total C U T A L O N G H E R E Please write your solutions on A4 paper and staple the above form to them. PLEASE WRITE YOUR NAME ON EVERY PAGE. Send your entry directly or through your school to : Simon Malham School of Mathematical and Computer Science Heriot Watt University Edinburgh EH14 4AS For further information on the competition, please see the Information Circular, which has been distributed to all secondary schools. Please contact the local organiser, whose name and address are given above, if you require a further copy. S5. In a wood there are more than 100 trees and all the trees have leaves on them. The number of trees in the wood is more than double the number of leaves on any one tree in the wood. Identify which of the following statements must be true: at least two trees have the same number of leaves on them; at least three trees have the same number of leaves on them; at least four trees have the same number of leaves on them. Explain your answer in each case. END OF PROBLEM SET 2 CLOSING DATE FOR RECEIPT OF SOLUTIONS : 18th February 2011 Look on the SMC web site: for information about Mathematical Challenge