Euclid Contest for The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Awards

Transcription

1 anadian Mathematics ompetition n activit of The entre for Education in Mathematics and omputing, Universit of Waterloo, Waterloo, Ontario Euclid ontest for The ENTRE for EUTION in MTHEMTIS and OMPUTING wards Wednesda, pril 4, 004.M.. Sponsors:.M.. Supporters: anadian Institute of ctuaries hartered ccountants Great West Life and London Life Sbase Inc. (Waterloo) inwhere Solutions Time: hours 004 Waterloo Mathematics Foundation alculators are permitted, provided the are non-programmable and without graphic displas. o not open this booklet until instructed to do so. The paper consists of 0 questions, each worth 0 marks. Parts of each question can be of two tpes. SHORT NSWER parts are worth marks each (questions -) or 3 marks each (questions 3-7). FULL SOLUTION parts are worth the remainder of the 0 marks for the question. Instructions for SHORT NSWER parts:. SHORT NSWER parts are indicated like this:.. Enter the answer in the appropriate bo in the answer booklet. For these questions, full marks will be given for a correct answer which is placed in the bo. Part marks will be awarded onl if relevant work is shown in the space provided in the answer booklet. Instructions for FULL SOLUTION parts:. FULL SOLUTION parts are indicated like this:.. Finished solutions must be written in the appropriate location in the answer booklet. Rough work should be done separatel. If ou require etra pages for our finished solutions, foolscap will be supplied b our supervising teacher. Insert these pages into our answer booklet. 3. Marks are awarded for completeness, clarit, and stle of presentation. correct solution poorl presented will not earn full marks. NOTE: t the completion of the contest, insert the information sheet inside the answer booklet.

2 NOTE:. Please read the instructions on the front cover of this booklet.. Place all answers in the answer booklet provided. 3. For questions marked, full marks will be given for a correct answer placed in the appropriate bo in the answer booklet. Marks ma be given for work shown if an incorrect answer is given. Students are strongl encouraged to show their work. 4. It is epected that all calculations and answers will be epressed as eact numbers such as 4π, + 7, etc., ecept where otherwise indicated.. (a) In the diagram, what is the area of figure EF? F 8 E In the diagram, is a rectangle with E = 5, E = 0 and F = 4. What is the length of F? 5 0 E 4 F (c) In the diagram, is a square of side length 6. Points E, F, G, and H are on,,, and, respectivel, so that the ratios E: E, F: F, G: G, and H: H are all equal to :. What is the area of EFGH? H E F G. (a) horizontal line has the same -intercept as the line 3 = 6. What is the equation of this horizontal line? In the diagram, line has equation =. Line is obtained b reflecting line in the -ais. Line is perpendicular to line. What is the slope of line? line line line

3 o (c) Three squares, each of side length, are drawn side b side in the first quadrant, as shown. Lines are drawn from the origin to P and Q. etermine, with eplanation, the length of. O P Q 3 3. (a) In an arithmetic sequence with five terms, the sum of the first two terms is and the sum of the last two terms is 8. What is the third term of this sequence? (n arithmetic sequence is a sequence in which each term after the first is obtained from the previous term b adding a constant. For eample, 3, 5, 7, 9, is an arithmetic sequence with five terms.) If = 4 and = 56, determine the two possible values of (a) Two fair dice, each having si faces numbered to 6, are thrown. What is the probabilit that the product of the two numbers on the top faces is divisible b 5? If f( )= +, g( )= a + b, and f( g( ) )= 9 3+, determine all possible ordered pairs ab, ( ) which satisf this relationship (a) If 6 =, determine the value of. In the diagram, the parabola with equation = + t intersects the -ais at points P and Q. lso, the line with equation = 3+ 3 intersects the parabola at points P and R. etermine the value of t and the area of triangle PQR. R P Q

4 6. (a) Lori has a loonie, three quarters, three dimes, three nickels, and five pennies. She wishes to purchase a to helicopter for eactl $.34. What is the maimum number of coins that she can use to make the purchase? (In anada, a loonie is worth $.00, a quarter is worth $0.5, a dime is worth $0.0, a nickel is worth $0.05, and a penn is worth $0.0.) igital images consist of a ver large number of equall spaced dots called piels. The resolution of an image is the number of piels/cm in each of the horizontal and vertical directions. Thus, an image with dimensions 0 cm b 5 cm and a resolution of 75 piels/cm has a total of ( 0 75) ( 5 75)= piels. If each of these dimensions was increased b n % and the resolution was decreased b n %, the image would have piels. etermine the value of n. 7. (a) In the diagram, =, = 7, = 8, and = 0 o. What is the value of? In the diagram, a drawer, PRST, that is cm high is in a long slot,, which is 5 cm high. The drawer is pulled out so that the 5 cm midpoint of its base rests at. The drawer is tilted so that the top back edge of the drawer touches the top of the slot. If the angle between the drawer and the slot is 0 o, determine the length of the drawer, to the nearest tenth of a centimetre. T P cm 0 Q S R 8. (a) If T = +, determine the values of b and c so that = T + bt + c for all 6 non-zero real numbers. If is a real number satisfing 3 + = 5, determine the eact value of 3 +.

5 ( ) of integers such that: 9. Kirk triplet is a triple,, z i) > z, ii) z is a prime number, and iii) there is a triangle with = =, =, and a point on such that = z and = 60. (a) Find the Kirk triplet with = 7 and z = 5. etermine all other Kirk triplets with z = 5. (c) etermine the Kirk triplet for which cos( ) is as close to 0.99 as possible. z ( ) of n integers satisfing the 0. Skolem sequence of order n is a sequence s, s,..., s n conditions: i) for ever k in { 3,,,..., n}, there eist eactl two elements s i and s j with si = sj = k, and ii) if si = sj = k with i < j, then j i = k. For eample, ( 4343,,,,,,, ) is a Skolem sequence of order 4. (a) List all Skolem sequences of order 4. etermine, with justification, all Skolem sequences of order 9 which satisf all of the following three conditions: I) s 3 =, II) s 8 = 8, and III) between an two equal even integers, there is eactl one odd integer. (c) Prove that there is no Skolem sequence of order n, if n is of the form 4k + or 4k + 3, where k is a non-negative integer.

6 PULITIONS 004 Euclid ontest (English) Students and parents who enjo solving problems for fun and recreation ma find the following publications of interest. The are an ecellent resource for enrichment, problem solving and contest preparation. opies of Previous anadian Mathematics ompetitions opies of previous contests and solutions are available at no cost in both English and French at Problems Problems Problems ooks Each volume is a collection of problems (multiple choice and full solution), grouped into 9 or more topics. Questions are selected from previous anadian Mathematics ompetition contests, and full solutions are provided for all questions. The price is $5. (vailable in English onl.) Volume 0 topics for students in Grades 9, 0, & French version of Volume is available Volume 3 over 35 problems and full solutions topics for senior high school students Volume 5 over 00 problems and full solutions 9 topics (different from Volume 3) for senior high school students Volume 7 topics for students in Grades 9 and 0 Volume 9 topics for students in Grades 7 and 8 Volume over 35 problems and full solutions 0 topics (different from Volume ) for students in Grades 9, 0, & Volume 4 over 35 problems and full solutions topics for students in Grades 7, 8, & 9 Volume 6 topics for students in Grades 7, 8, & 9 Volume 8 over 00 problems and full solutions 0 topics for students in Grades and Orders should be addressed to: anadian Mathematics ompetition Facult of Mathematics, Room 58 Universit of Waterloo Waterloo, ON NL 3G Include our name, address (with postal code), and telephone number. heques or mone orders in anadian funds should be made paable to "entre for Education in Mathematics and omputing". In anada, add $3.00 for the first item ordered for shipping and handling, plus $.00 for each subsequent item. No Provincial Sales Ta is required, but 7% GST must be added. Orders outside of anada ONLY, add $0.00 for the first item ordered for shipping and handling, plus $.00 for each subsequent item. Prices for these publications will remain in effect until September, 004. NOTE: ll publications are protected b copright. It is unlawful to make copies without the prior written permission of the Waterloo Mathematics Foundation.