2006 ALABAMA MATHEMATIC CONTEST (2)
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Muhammad Yusuf
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11. If m and n are required to be integers, how many solutions (m, n) are there to the pair of
conditions 5n − 3m = 15
and n2 + m2 ≤ 16 ?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
12. If c = logx b and d = logx2 (b2), find d in terms of c .
(A) 2c (B) c−1 (C) c (D) c2 (E) c/2
13. If the arithmetic mean of two
positive numbers is A and the geometric mean is B, what are the two numbers ?
14. Suppose x and y are two unequal numbers.
Their sum, difference, and product are denoted by S, D, and P, respectively. We know that S/D = 7 and P/D = 24. What is the value of P ?
(A) 42 (B) 31 (C) 28 (D) 48 (E) 14
15. If one and one-half teams of men
require one and one-half days to shovel one and one-half truckloads of sand,
how long will it take one team to shovel one truckload ?
(A) 3
days (B) 2 days (C) 2/3 days (D) 1 day (E) 1 and 1/2 days
16. In the expansion of (x − √x − 1)7, what is the coefficient of x5/2 ?
(A) 231
(B) 224 (C) 14 (D) 266 (E) 56
17. What is the smallest positive
integer n for which 45n is a perfect cube of an integer ?
(A) 75 (B) 2025 (C) 625 (D) 55 (E) 245
18. Three positive numbers a, b, and c form an arithmetic progression. Oddly enough, if
you increase a by 1 you get a geometric
progression. Even more oddly, if you increase c by 2
you also get a geometric progression. What is the value of b ?
(A) 8 (B) 9 (C) 10 (D) 12 (E) 18
19. Simplify
(A) 2 tan x (B) cos x − sin x (C) 2 cot x (D) cos x + sin x (E) None of these
20. What is the value of
(A) 1/9 (B) 43046721 (C) 1/3 (D) 6561
(E) √3/3
21. Write the repeating decimal 0.35353535 . . . as a fraction.
(A) 353/909 (B) 7/18 (C) 353/990 (D) 353/999 (E) 35/99
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