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1、Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing University of Waterloo Waterloo Ontario Euclid Contest Tuesday April 7 2009 C M C Sponsors Chartered Accountants C M C Supporter Time 21 2 hoursc 2009 Centre for Education in Mathematics and Computi
2、ng Calculators are permitted provided they are non programmable and without graphic displays Do not open this booklet until instructed to do so The paper consists of 10 questions each worth 10 marks Parts of each question can be of two types SHORT ANSWER parts are worth 3 marks each FULL SOLUTION pa
3、rts are worth the remainder of the 10 marks for the question Instructions for SHORT ANSWER parts 1 SHORT ANSWER parts are indicated like this 2 Enter the answer in the appropriate box in the answer booklet For these questions full marks will be given for a correct answer which is placed in the box P
4、art marks will be awarded only if relevant work is shown in the space provided in the answer booklet Instructions for FULL SOLUTION parts 1 FULL SOLUTION parts are indicated like this 2 Finished solutions must be written in the appropriate location in the answer booklet Rough work should be done sep
5、arately If you require extra pages for your fi nished solutions foolscap will be supplied by your supervising teacher Insert these pages into your answer booklet Be sure to write your name school name and question number on any inserted pages 3 Marks are awarded for completeness clarity and style of
6、 presentation A correct solution poorly presented will not earn full marks NOTE At the completion of the Contest insert the information sheet inside the answer booklet The names of some top scoring students will be published in the Euclid Results on our Web site http www cemc uwaterloo ca NOTES 1 Pl
7、ease read the instructions on the front cover of this booklet 2 Write all answers in the answer booklet provided 3 For questions marked full marks will be given for a correct answer placed in the appropriate box in the answer booklet If an incorrect answer is given marks may be given for work shown
8、Students are strongly encouraged to show their work 4 All calculations and answers should be expressed as exact numbers such as 4 2 7 etc rather than as 12 566 or 4 646 except where otherwise indicated A Note about Bubbling Please make sure that you have correctly coded your name date of birth grade
9、 and sex on the Student Information Form and that you have answered the question about eligibility A Note about Writing Solutions For each problem marked a full solution is required The solutions that you provide in the answer booklet should be well organized and contain mathematical statements and
10、words of explanation when appropriate Working out some of the details in rough on a separate piece of paper before writing your fi nished solution is a good idea Your fi nal solution should be written so that the marker can understand your approach to the problem and all of the mathematical steps of
11、 your solution 1 a A line has equation 6x 3y 21 0 What is the slope of the line b A line with a slope of 3 passes through the points 1 0 and 5 c What is the value of c c The point k k lies on the line segment AB shown in the diagram Determine the value of k y x B 8 4 A 0 4 O 2 a What is the sum of t
12、he two numbers that satisfy the equation x2 6x 7 0 b What is the product of the two numbers that satisfy the equation 5x2 20 0 c Determine the average of the numbers that satisfy the equation x3 6x2 5x 0 3 a In the diagram AB AC AD BD and CAE is a straight line segment that is perpendicular to BD Wh
13、at is the measure of CDB C BD A E b In rectangle ABCD F is on diagonal BD so that AF is perpendicular to BD Also BC 30 CD 40 and AF x Determine the value of x AB CD F x 4 a In an arithmetic sequence the fi rst term is 1 and the last term is 19 The sum of all the terms in the sequence is 70 How many
14、terms does the sequence have 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 is an arithmetic sequence with four terms b Suppose that a x b x 3 2 x 6 for all values of x Determine a and b 5 a In the
15、diagram 4ABC is isosceles with AC BC 7 Point D is on AB with CDA 60 AD 8 and CD 3 Determine the length of BD AB D C 60 8 3 77 b In the diagram 4ABC is right angled at C Also 2sinB 3tanA Determine the measure of angle A C A B b a c 6 a An integer n with 100 n 999 is chosen at random What is the proba
16、bility that the sum of the digits of n is 24 b Alice drove from town E to town F at a constant speed of 60 km h Bob drove from F to E along the same road also at a constant speed They started their journeys at the same time and passed each other at point G EF G Alice Bob Alice drove from G to F in 45 minutes Bob drove from G to E in 20 minutes Determine Bob s constant speed 7 a The parabola y x2 2x 4 is translated p units to the right and q units down The x intercepts of the resulting parabola a
