Secondary Mathematical Challenges

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Aubrey Caldwell
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1 Secondary Mathematical Challenges Welcome to the second round of the Scottish Secondary Mathematical Challenges. This package contains This Welcome Page (including Section Information) Round 2 Questions In , the name of the Section Organiser is not on the question paper. Their details are on the website but are repeated here for convenience. Please take great care to use the correct one. Section 1 Aberdeen City; Aberdeenshire; Highland; Moray; Orkney Islands; Shetland Islands; Western Isles Dr William Turner (w.turner@abdn.ac.uk) Mathematical Challenge Department of Mathematical Sciences, University of Aberdeen, Aberdeen A24 3UE Section 2 Angus; Clackmannanshire; Dundee City; Falkirk; Fife; Perth & Kinross; Stirling Dr Jean Reinaud (jnr1@st-andrews.ac.uk) Mathematical Challenge Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS Section 3 East Lothian; Edinburgh City; Midlothian; Scottish orders; West Lothian Dr Lotte Hollands (l.hollands@hw.ac.uk) Mathematical Challenge Department of Mathematics, Heriot Watt University, Edinburgh EH14 4AS Section 4 Argyll & ute; Dumfries & Galloway; East Ayrshire; East Dunbartonshire; East Renfrewshire; Glasgow City; Inverclyde; North Ayrshire; North Lanarkshire; Renfrewshire; South Ayrshire; South Lanarkshire; West Dunbartonshire Dr Chris Athorne Department of Mathematics, University of Glasgow, University Gardens Glasgow G12 8QW The competition timetable for is as follows: Set Last date for receipt of questions by schools Last date for receipt of solutions from pupils I Friday 25 August 2017 Friday 29 September 2017 II Friday 24 November 2017 Friday 23 February 2018 If there are organisational difficulties you may contact me, ill Richardson, on: wpr3145@gmail.com ooks of past questions are still available but it seems unlikely that any more will be printed so questions and solutions for can be accessed at:

2 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, The Maxwell Foundation, 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, Heriot Watt, St Andrews, Stirling, Strathclyde and to Gryffe Academy, Kelvinside Academy and Northfield Academy. J1. Junior Division: Problems 2 Trains on the Glasgow Subway depart every 4 minutes, and a complete circuit takes 24 minutes. Ewan sets off at 8.30 am on a train round in one direction at the same time as another train leaves in the opposite direction. How many trains will he pass on a complete circuit back to his starting station? (Do not count trains at the start or end station.) J2. Some people think it is unlucky if the 13th day of a month falls on a Friday. Show that in every calendar year (non-leap or leap) there will always be at least one such unlucky Friday but that there can be no more than three. J3. Three energy saving improvements are advertised to save 25%, 55% and 20% of the energy used. A homeowner makes these three improvements in succession. What overall percentage saving can be expected? J4. A strange announcement was made on the radio about a local election with three candidates: Mrs Allan, Mr axter and Ms Campbell. Mrs Allan beat Mr axter by one eighth of the total votes cast. Mr axter beat Ms Campbell by a seventh of the total votes cast. The votes cast for Mrs Allan was 350 fewer than 3 times Ms Campbell's votes. How many votes did each candidate get? SEE OVER FOR QUESTION J5.

3 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 through your school to the section organsier. 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. J5. A trapezium ACD is split into four identical trapezia as shown below. C A D Given that A has length 6 cm, find the area of ACD. CLOSING DATE FOR RECEIPT OF SOLUTIONS : END OF PROLEM SET 2 23 February 2018 Look on the SMC web site: for information about Mathematical Challenge

4 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, The Maxwell Foundation, 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, Heriot Watt, St Andrews, Stirling, Strathclyde and to Gryffe Academy, Kelvinside Academy and Northfield Academy. Middle Division: Problems 2 M1. A strange announcement was made on the radio about a local election with three candidates: Mrs Allan, Mr axter and Ms Campbell. Mrs Allan beat Mr axter by one eighth of the total votes cast. Mr axter beat Ms Campbell by a seventh of the total votes cast. The votes cast for Mrs Allan was 350 fewer than 3 times Ms Campbell's votes. How many votes did each candidate get? M2. A trapezium ACD is split into four identical trapezia as shown below. C A D Given that A has length 6 cm, find the area of ACD. M3. Red, yellow and blue counters are placed on a board as shown, and they race to the finish (F) by moving up one square at a time. The moves are determined by picking a bead at random from a bag containing one red bead, two yellow beads and three blue beads. After the colour of the bead which has been drawn is noted, the bead is returned to the bag before the next bead is picked. The race is over as soon as one of the counters lands on the square marked F. Find the probability of winning for each of the counters. F F F R Y M4. Goliath and David play a game in which there are no ties. Each player is equally likely to win each game. The first player to win 4 games becomes the champion, and no further games are played. Goliath wins the first two games. What is the probability that David becomes the champion? SEE OVER FOR QUESTION M5.

5 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 through your school to the section organsier. 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. Let n be a three-digit number and let m be the number obtained by reversing the order of the digits in n. Suppose that m does not equal n and that n + m and n m are both divisible by 7. Find all such pairs n and m. END OF PROLEM SET 2 CLOSING DATE FOR RECEIPT OF SOLUTIONS : 23 February 2018 Look on the SMC web site: for information about Mathematical Challenge

6 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, The Maxwell Foundation, 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, Heriot Watt, St Andrews, Stirling, Strathclyde and to Gryffe Academy, Kelvinside Academy and Northfield Academy. Senior Division: Problems 2 S1. Goliath and David play a game in which there are no ties. Each player is equally likely to win each game. The first player to win 4 games becomes the champion, and no further games are played. Goliath wins the first two games. What is the probability that David becomes the champion? S2. Let n be a three-digit number and let m be the number obtained by reversing the order of the digits in n. Suppose that m does not equal n and that n + m and n m are both divisible by 7. Find all such pairs n and m. S3. ACD is a square. Points P and Q lie within the square such that AP, PQ and QC are all the same length and AP is parallel to QC. Determine the minimum possible size of DAP. D P C Q A S4. Determine all values of x for which ( x ) log 10 x = 100. SEE OVER FOR QUESTION S5.

7 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 through your school to the section organsier. 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 quadrilateral PQRS, the sides PQ and SR are parallel, and the diagonal QS bisects angle PQR. Let X be the point of intersection of the diagonals PR and QS. PX Prove that. XR = PQ QR In a triangle AC the lengths of all three sides are positive integers. The point M lies on the side C so that AM is the internal bisector of the angle AC. Also, M = 2 and MC = 3. What are the possible lengths of the sides of the triangle AC? END OF PROLEM SET 2 CLOSING DATE FOR RECEIPT OF SOLUTIONS : 23 February 2018 Look on the SMC web site: for information about Mathematical Challenge

POLICE SCOTLAND PROFESSIONAL STANDARDS DEPARTMENT NATIONAL STATISTICAL PERFORMANCE REPORT

NOT PROTECTIVELY MARKED SIONAL STANDARD POLICE SCOTLAND PROFESSIONAL STANDARDS DEPARTMENT NATIONAL STATISTICAL PERFORMANCE REPORT Covering the period: April 2016 March 2017 NOT PROTECTIVELY MARKED Total