Math Circles 9 / 10 Contest Preparation I

Transcription

1 Math Circles 9 / 10 Contest Preparation I Centre for Education in Mathematics and Computing CEMC February 4, 2015

2 Agenda 1 Warm-up Problem 2 Contest Information 3 Contest Format 4 Some General Comments About Preparing Through Examples 5 Working on Problems

3 WARM-UP: Hockey is Back! Four teams A, B, C, and D competed against each other. Unlike the NHL, games in this league can end in a tie. The following table summarizes the results of the games: Team Wins Ties Losses Goals For Goals Against A B C D In the final game, B played C and the score was 1-1. a) Which team won in each of the six matches? b) What were the scores in the matches that B played? c) What were the scores in the rest of the matches?

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5 Contest Information Pascal for students in grade 9 or lower Cayley for students in grade 10 or lower Fermat for students in grade 11 or lower Tuesday, February 24, 2015 Deadline for entry: February 10, 2015

6 The Contest Format From the instructions: This is a multiple-choice test. Each question is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct.

7 The Contest Format 60 minutes 150 total marks 10 questions in A part worth 5 marks each 10 questions in B part worth 6 marks each 5 questions in C part worth 8 marks each Each unanswered question is worth 2 marks each to a maximum of 10 unanswered questions. (non-programmable) calculator permitted

8 The Contest: Should I Guess? 150 total marks (A) 10 5 (B) 10 6 (C) 5 8 Unanswered questions 2 marks each to a maximum of 10 blank

9 The Contest: Should I Guess? 150 total marks (A) 10 5 (B) 10 6 (C) 5 8 Unanswered questions 2 marks each to a maximum of 10 blank Score With Random Guessing: Part A 10 right, Part B 8 right 2 wrong, Part C 2 right 3 wrong Score = 114

10 The Contest: Should I Guess? 150 total marks (A) 10 5 (B) 10 6 (C) 5 8 Unanswered questions 2 marks each to a maximum of 10 blank Score With Random Guessing: Part A 10 right, Part B 8 right 2 wrong, Part C 2 right 3 wrong Score = 114 Score With No Guessing: Part A 10 right, Part B 8 right 2 blank, Part C 2 right 3 blank Score = 124

11 The Contest: Should I Guess? 150 total marks (A) 10 5 (B) 10 6 (C) 5 8 Unanswered questions 2 marks each to a maximum of 10 blank Score With Random Guessing: Part A 10 right, Part B 8 right 2 wrong, Part C 2 right 3 wrong Score = 114 Score With No Guessing: Part A 10 right, Part B 8 right 2 blank, Part C 2 right 3 blank Score = 124 There is a difference between a random guess and an educated guess.

12 Contest Preparation Most of the problems that we will look at have been taken from past Pascal and Cayley contests. The key to success on any math contest is. Writing a Math contest is different than solving math problems in general. The contest is timed! By doing many old contests you will pick up some tricks and techniques which will help you to write a good contest.

13 Contest Resources - eworkshops Let s try: Pascal, Cayley, Fermat eworkshop

14 Contest Resources - Old Contests Let s try: Pascal, Cayley, Fermat Old Contests

15 Can we eliminate any answers? From PCF EWorkshop Practice Cayley #2 Question 1 If x = 2 and y = 5 then (x y)(x + y) equals (A) 40 (B) 21 (C) 0 (D) -21 (E) -49

16 Be sure to get the easier ones correct. Try these! (Pascal #3) If a = 4, b = 5, and c = 9, then the value of (a + b c) + (a b + c) is (A) 8 (B) 0 (C) -10 (D) 10 (E) 16 (Pascal #4) The number of odd integers between 19 4 and 43 2 is (A) 8 (B) 9 (C) 10 (D) 16 (E) 17 (Pascal #6) The ratio of the number of red marbles to the number of green marbles in a container is 1 : 3. If there are 48 marbles altogether, then the number of green marbles is (A) 12 (B) 16 (C) 36 (D) 32 (E) 24

17 Can we eliminate answers? (Pascal #5) In a magic square, the sum of the entries in each row, in each column, and in each diagonal are all equal N 17 In the magic square shown, the value of N is (A) 5 (B) 13 (C) 7 (D) 16 (E) 3

18 How can we think about this one? (Pascal #19) During a high school drama rehearsal, 15 girls left. Twice as many boys as girls remained, Later, 45 boys departed, leaving five times as many girls as boys. Before anyone left the rehearsal, there were (A) 10 more girls than boys (C) 25 more boys than girls (E) 20 more girls than boys (B) 20 more boys than girls (D) 10 more boys than girls

19 How can we think about this one? (Pascal #16) The number of integers between 2 and 50 that can be written in the form x y, where x and y are positive integers and y 1, is (A) 8 (B) 7 (C) 10 (D) 6 (E) 9

20 How can we think about this one? (Pascal #14) The eight digits 6, 5, 5, 4, 4, 3, 2, 1 are used to form two three-digit numbers and one two-digit number. The largest possible sum of these numbers is (A) 1119 (B) 1713 (C) 1218 (D) 30 (E) 1236

21 How can we think about this one? (Pascal #18) Each of the numbers from 1 to 9 is placed, one per circle, into the pattern shown. The sums along each of the four sides are equal. The number of possible entries for the middle circle is (A) 2 (B) 9 (C) 1 (D) 5 (E) 3

22 Earlier generally means easier! (Pascal #7) The diagram shows a square PQRS and two equilateral triangles RSU and PST. PQ has length 1. The length of TU is (A) 3 2 (B) (C) 3 (D) 5 1 (E) 6 1 2

23 Earlier generally means easier! (Pascal #21) In the Big 15 Lottery, each of the numbers from 1000 to 9999 inclusive is printed, one to a ticket. A number is considered a winner if its hundreds digit is 8, its tens digit is 6, and it is divisible by 15. The number of winning tickets is (A) 3 (B) 9 (C) 2 (D) 15 (E) 6

24 Earlier generally means easier! (Pascal #23) In the sequence 6, 14, 8, 6,, every term after the second is the difference of the preceding two terms in reverse order. For example, 8 = 14 6 and 6 = The sum of the first 2000 terms of the sequence is (A) 28 (B) 8 (C) 22 (D) 0 (E) 20

25 Earlier generally means easier! (Pascal #24) On planet Binad, a Bank Cash mashine uses eight ON-OFF switches numbered 1 to 8 to allow deposits to and withdrawals from an account. All the switches start out OFF. When switch n is flipped ON, the balance of the account changes by ( 2) n dollars. For example, if switches 4 and 7 are flipped ON and the other switches are left OFF, a withdrawal of 112 dollars is made. The number of switches that must be turned ON to deposit 114 dollars is (A) 4 (B) 6 (C) 5 (D) 3 (E) 7

26 Work Time Work on Contest Preparation 1. Work independently at first. Consult your neighbours after you have attempted all the problems. Solutions will be posted online next week. Questions for me?

27 Answers Only to Contest Preparation 1 1. D 2. A 3. A 4. B 5. E 6. C 7. B 8. B 9. A 10. A

28 Answers Only to Contest Preparation 1 1. D 2. A 3. A 4. B 5. E 6. C 7. B 8. B 9. A 10. A 11. C 12. E 13. A 14. E 15. D 16. E 17. C 18. A 19. B 20. D

29 Answers Only to Contest Preparation 1 1. D 2. A 3. A 4. B 5. E 6. C 7. B 8. B 9. A 10. A 11. C 12. E 13. A 14. E 15. D 16. E 17. C 18. A 19. B 20. D 21. B 22. D 23. B 24. B 25. C Questions for me? troy.vasiga@uwaterloo.ca