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1、The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc uwaterloo ca Euclid Contest Wednesday April 17 2013 in North America and South America Thursday April 18 2013 outside of North America and South America c 2013 University of Waterloo Do not open this booklet until instructed to do so Time 21
2、 2 hoursNumber of questions 10 Calculators are permitted providedEach question is worth 10 marks they are non programmable and without graphic displays Parts of each question can be of two types 1 SHORT ANSWER parts indicated by worth 3 marks each full marks given for a correct answer which is place
3、d in the box part marks awarded only if relevant work is shown in the space provided 2 FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness clarity and style of presentat
4、ion a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED Extra paper for your fi nished solutions supplied by your supervising teacher must be inserted into your answer booklet Write your name school name and question number on any inserted pa
5、ges Express calculations and answers as exact numbers such as 1 and 2 etc rather than as 4 14 or 1 41 except where otherwise indicated Do not discuss the problems or solutions from this contest online for the next 48 hours The name grade school and location and score range of some top scoring studen
6、ts will be published on our website http www cemc uwaterloo ca In addition the name grade school and location and score of some top scoring students may be shared with other mathematical organizations for other recognition opportunities TIPS 1 Please read the instructions on the front cover of this
7、booklet 2 Write all answers in the answer booklet provided 3 For questions marked place your answer in the appropriate box in the answer booklet and show your work 4 For questions marked provide a well organized solution in the answer booklet Use mathematical statements and words to explain all of t
8、he steps of your solution Work out some details in rough on a separate piece of paper before writing your fi nished solution 5 Diagrams are not drawn to scale They are intended as aids only A Note about Bubbling Please make sure that you have correctly coded your name date of birth grade and sex on
9、the Student Information Form and that you have answered the question about eligibility 1 a What is the smallest positive integer x for which 113 x is an integer b The average of 3 and 11 is a The average of a and b is 11 What is the value of b c Charlie is 30 years older than his daughter Bella Char
10、lie is also six times as old as Bella Determine Charlie s age 2 a If 21 x 7 y with x 6 0 and y 6 0 what is the value of x y b For which positive integer n are both 1 n 1 0 2013 and 0 2013 1 n true c In the diagram H is on side BC of 4ABC so that AH is perpendicular to BC Also AB 10 AH 8 and the area
11、 of 4ABC is 84 Determine the perimeter of 4ABC A BC H 3 a In the Fibonacci sequence 1 1 2 3 5 each term after the second is the sum of the previous two terms How many of the fi rst 100 terms of the Fibonacci sequence are odd b In an arithmetic sequence the sum of the fi rst and third terms is 6 and
12、the sum of the second and fourth terms is 20 Determine the tenth term in the sequence An arithmetic sequence is a sequence in which each term after the fi rst is obtained from the previous term by adding a constant For example 3 5 7 9 are the fi rst four terms of an arithmetic sequence 4 a How many
13、positive integers less than 1000 have only odd digits b Determine all ordered pairs a b that satisfy the following system of equations a b 16 4 7 1 a 1 b 5 a Tanner has two identical dice Each die has six faces which are numbered 2 3 5 7 11 13 When Tanner rolls the two dice what is the probability t
14、hat the sum of the numbers on the top faces is a prime number b In the diagram V is the vertex of the parabola with equation y x2 4x 1 Also A and B are the points of intersection of the parabola and the line with equation y x 1 Determine the value of AV 2 BV 2 AB2 y x A V B 6 a In the diagram ABC is
15、 a quarter of a circular pizza with centre A and radius 20 cm The piece of pizza is placed on a circular pan with A B and C touching the circumference of the pan as shown What fraction of the pan is covered by the piece of pizza AB C b The deck AB of a sailboat is 8 m long Rope extends at an angle o
16、f 60 from A to the top M of the mast of the boat More rope extends at an angle of from B to a point P that is 2 m below M as shown Determine the height MF of the mast in terms of 60 M AB F P 7 a If 1 cosx tanx 3 what is the numerical value of sinx b Determine all linear functions f x ax b such that if g x f 1 x for all values of x then f x g x 44 for all values of x Note f 1is the inverse function of f 8 a Determine all pairs a b of positive integers for which a3 2ab 2013 b Determine all real va
