Cayley Contest (Grade 10) Wednesday, February 23, 2005

Transcription

1 anadian Mathematics ompetition n activity of the entre for Education in Mathematics and omputing, University of Waterloo, Waterloo, Ontario ayley ontest (Grade 10) Wednesday, February 23, 2005.M.. Sponsors:.M.. Supporters: anadian Institute of ctuaries hartered ccountants Great West Life and London Life Sybase inywhere Solutions Time: 60 minutes c 2004 Waterloo Mathematics Foundation alculators are permitted. Instructions 1. o not open the ontest booklet until you are told to do so. 2. You may use rulers, compasses and paper for rough work. 3. e sure that you understand the coding system for your response form. If you are not sure, ask your teacher to clarify it. ll coding must be done with a pencil, preferably H. Fill in circles completely. 4. On your response form, print your school name, city/town, and province in the box in the upper left corner. 5. e certain that you code your name, age, sex, grade, and the ontest you are writing in the response form. Only those who do so can be counted as official contestants. 6. This is a multiple-choice test. Each question is followed by five possible answers marked,,,, and E. Only one of these is correct. fter making your choice, fill in the appropriate circle on the response form. 7. Scoring: Each correct answer is worth 5 in Part, 6 in Part, and 8 in Part. There is no penalty for an incorrect answer. Each unanswered question is worth 2, to a maximum of 10 unanswered questions. 8. iagrams are not drawn to scale. They are intended as aids only. 9. When your supervisor tells you to begin, you will have sixty minutes of working time.

2 Scoring: There is no penalty for an incorrect answer. Each unanswered question is worth 2, to a maximum of 10 unanswered questions. Part : Each correct answer is worth The expression a a 2 + a a 4 is equal to () 10a () 0 () 4a 2 () 4a + 2 (E) 2a 2. The value of ( 4 5 )( 5 6 )( 6 7 )( 7 8 )( 8 9 ) is () 4 9 () 1 () 6 7 () 36 (E) When 45 is divided by 7, the remainder is 3. What is the remainder when 70 is divided by 17? 4. If () 1 () 12 () 15 () 2 (E) 11 3 x + 10 = 1, then x equals 2x () 1 2 () 10 () 4 () 2 (E) 8 5. teacher writes five different possible values for ( ) 3 on the board and asks her class to decide which is correct. The correct value is () 1 () 8 () () 216 (E) Last week, a charity fundraiser had 8 volunteers who each worked 40 hours and who each raised $18 per hour. This week, 12 volunteers, each working 32 hours, raised the same total amount of money. How much did each volunteer raise per hour this week? () $9 () $12 () $15 () $21 (E) $24 7. In the diagram, the line segment has slope 3 2. The value of b is () 10 () 12 () 6 () 16 (E) 20 y (0,b) O (8,0) x 8. Jack went running last Saturday morning. He ran the first 12 km at 12 km/h and the second 12 km at 6 km/h. Jill ran the same route at a constant speed, and took the same length of time as Jack. Jill s speed in km/h was () 8 () 9 () 6 () 12 (E) 24

3 9. is a rectangle, with M the midpoint of and N the midpoint of. If M = 4 and N = 5, what percent of the area of the rectangle is shaded? () 70 () 78 () 80 M () 87.5 (E) 75 N 10. In the diagram, P T is parallel to QR. What is the size of P QR? P x o T 2x o () 116 () 168 () 138 () 144 (E) 122 Q 128 o R Part : Each correct answer is worth uring a football game, Matt kicked the ball three times. His longest kick was 43 metres and the three kicks averaged 37 metres. If the other two kicks were the same length, the distance, in metres, that each travelled was () 31 () 37 () 35 () 34 (E) The lines y = 2x + 8 and y = 1 2x 2 meet at (4, 0), as shown. The area of the triangle formed by these two lines and the line x = 2 is () 15 () 27 () 30 () 36 (E) 45 y y = 1 2 x 2 (4,0) x y = 2x + 8 x = m track is constructed so that the points,,, and divide the track into four segments of equal length. The Start is half-way between and. ndrew begins at the Start and walks at a steady rate of 1.4 m/s in a counter-clockwise direction. fter exactly 30 minutes, to what point will ndrew be closest? Start () () () () (E) Start 14. If x is a positive integer less than 100, how many values of x make x an integer? () 6 () 7 () 8 () 9 (E) 10

4 15. Starting with the 2 in the centre, the number 2005 can be formed by moving from circle to circle only if the two circles are touching. How many different paths can be followed to form 2005? () 36 () 24 () 12 () 18 (E) The non-negative difference between two numbers a and b is a b or b a, whichever is greater than or equal to 0. For example, the non-negative difference between 24 and 64 is 40. In the sequence 88, 24, 64, 40, 24,..., each number after the second is obtained by finding the non-negative difference between the previous 2 numbers. The sum of the first 100 numbers in this sequence is () 496 () 760 () 752 () 776 (E) is a googol equals () 100 googol () 3 googol () googol googol () googol 2 (E) googol Harry the Hamster is put in a maze, and he starts at point S. The paths are such that Harry can move forward only in the direction of the arrows. t any junction, he is equally likely to choose any of the forward paths. What is the probability that Harry ends up at? () 2 3 () () () 1 3 (E) 1 4 S 19. In the diagram, = 13 cm, = 20 cm, and = 5 cm. The length of, to the nearest tenth of a centimetre, is () 24.2 () 20.6 () cm 20 cm () 23.4 (E) cm 20. There are 81 cars in the M parking lot, which are all curas, eetles, or amrys. There are half as many curas as eetles. The number of amrys is 80% of the number of curas and eetles together. How many of the 81 cars are eetles? () 36 () 30 () 45 () 51 (E) 66

5 Part : Each correct answer is worth In Yacleyland, the unit of money used is called the Yacley. There are only two denominations of paper money: the 17 Yacley bill and the 5 Yacley bill. How many different combinations of these bills total 453 Yacley? () 3 () 4 () 5 () 6 (E) In the diagram, O is a quarter circle of radius 10 and P QRO is a rectangle of perimeter 26. The perimeter of the shaded region is () 7 + 5π () π () π () π (E) π Q R P O 23. t 12:00 noon, nna and ill left home and walked in the same direction. nna walked at 4 km/h and ill walked at 3 km/h. t 12:15 their dog exter, running at 6 km/h, left home to run after them. The dog ran until it caught up to nna, then it ran back to ill. (In his excitement, exter lost no time in turning around once he reached nna.) t what time did ill meet exter on exter s way back? () 1:00 p.m. () 1:15 p.m. () 12:45 p.m. () 1:05 p.m. (E) 12:50 p.m. 24. The base of a triangular piece of paper is 12 cm long. The paper is folded down over the base, with the crease E parallel to the base of the paper. The area of the triangle that projects below the base is 16% that of the area of the triangle. The length of E, in cm, is E () 9.6 () 8.4 () 7.2 () 4.8 (E) cm 25. The positive integers a, b and c satisfy 1 a b 2 = 1 c 2. The sum of all possible values of a 100 is () 315 () 615 () 680 () 555 (E) 620

6 2005 ayley ontest (English) anadian Mathematics ompetition For students... Thank you for writing the 2005 ayley ontest! In 2004, more than students around the world registered to write the Pascal, ayley and Fermat ontests. Encourage your teacher to register you for Galois ontest which will be written on pril 20, Visit our website to find For teachers... More information about the Galois ontest Free copies of past ontests Workshops to help you prepare for future ontests Information about our publications for math enrichment and ontest preparation Information about careers in math Visit our website to Register your students for the Fryer, Galois and Hypatia ontests which will be written on pril 20, 2005 Learn about workshops and resources we offer for teachers Find your school results