Let f(a ; b) be the sum of all the positive integers between a and b inclusive. For example,
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1 Question 1: Let f(a ; b) be the sum of all the positive integers between a and b inclusive. For example, f(1 ; 5) = = 15. Determine the value of f(13 ; 53) N = 41 terms = S = n ( first term + last term) 2 = 41 ( ) 2 = 41 ( 66 ) / 2 = 41 (33) = 1353
2 Question 2: In how many different ways can these people be arranged in a line next to each other for the photograph if the bride and groom must be together? Couple can be seen as one person = 24 Since bride and groom can switch sides in any of the above arrangements = times = 48 ways
3 Question 3: Three gears mesh together as shown. In the time gear A makes six revolutions, how many revolutions does gear C make? 12:9. So 6 revolutions will be 72 clicks, which will revolve C 8 times.
4 Question 4: How many different triangles (of all sizes) are in this figure? 16 triangles
5 Question 5: The figure is a combination of four semi-circles, each with a radius of 3 cm. What is the area of the figure? 3 2 = 6 cm Forms a square 6 by 6 = 36
6 Question 6: When is evaluated, what is the sum of the digits? = (47 nines 0 0 1) results in = 424
7 Question 7: Let x, y be two positive integers. Suppose x 2 = 40 + y 2. Find the largest possible value of (x - y). x 2 - y 2 = 40 (x y)(x + y) = 40 Only possible terms that can produce integers for x and y are x y = 4 x + y = 10 2x = 14 x = 7, thus y = 3 Other combinations like 8 and 5 or 1 and 40 result in an odd sum, which will not give us integers for x and y.
8 Question 8: A spiral staircase, with a constant slope, circles around the outside of the water tower exactly twice according to the diagram below. What is the length of this staircase? = /π C= 2π(12/π)= Therefore, total length is = 50 m
9 Question 9: What is the largest prime factor of ? = 7 99 ( ) = 7 99 (57) = 7 99 (3 29) Therefore the largest prime is 29.
10 Relay 1: 1. A is the product of the following pattern below. (1 + 1) (1 +1/2 ) (1 +1/3 ) (1 +1/4 ) (1 +1/5 ) (1 + 1/100 ) Write the value of A in the box #1 of the Relay Answer Sheet. 2. When A 7, B is the remainder. Write the value of B in Box # 2 of the Relay Answer Sheet. 3. If (x + B)(x + p) = x 2 + Cx 12 for some integers p and C. Write the value of C in Box # 3 of the Relay Answer Sheet. 4. [ 5 / (3 2 C)] C = D Write the value of D in Box # 4 of the Relay Answer Sheet.
11 Relay 2: 1. (3 12 )(4 5 ) = 6 A 9 Write the value of A in Box # 1 of the Relay Answer Sheet. 2. B is the A th number that will appear in the following sequence. 2; 5; 10; 17; 26; 37;... Write the value of B in Box # 2 of the Relay Answer Sheet. 3. C is the units digit of 2 B Write the value of C in Box # 3 of the Relay Answer Sheet. 4. x 2 Cx 48 = 0. D is the largest zero(x-intercept). Write the value of D in Box # 4 of the Relay Answer Sheet.
12 Relay 3: 1. Set G has twice the number of elements as set H. A = n ( G H ) G x - 5 x 2x + 1 H Write the value of A in Box # 1 of the Relay Answer Sheet. 2. B is the remainder when 2011 A is divided by 100. Write the value of B in Box # 2 of the Relay Answer Sheet. 3. C = (B 4) 3. (4 + B) (B + 9) Write the value of C in Box # 3 of the Relay Answer Sheet. 4. D is the sum of the first 80 terms of the following sequence: 5, B, 3, B, 1, B, 3, B, 5, B, 3, B, 1, B, 3, B, 5, B, 1, B, Write the value of D in Box # 4 of the Relay Answer Sheet.
13 Relay # 1 - Answers A 101 B 3 C -1 D 2 Relay # 1 - Answer Sheet TEAM # School: A B C D Regular points (max. 5) + Bonus Points (max. 6) = Total Points Proctors Initials:
14 Relay # 2 - Answers A 10 B 101 C 2 D 8 Relay # 2 - Answer Sheet TEAM # School: A B C D Regular points (max. 5) + Bonus Points (max. 10) = Total Points Proctors Initials:
15 Relay # 3 - Answers A 11 B 1 C -2 D 40 Relay # 3 - Answer Sheet TEAM # School: A B C D Regular points (max. 5) + Bonus Points (max. 10) = Total Points Proctors Initials:
16 Answers
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