2011 Alabama Mathematics Contest : Algebra With Trigonometry
5:49 AM
Muhammad Yusuf
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1. Suppose x is a complex number satisfying the equation x + 1/x = 1. What is the value of x3 + 1/x3 ?
(A) −2 (B) −1 (C) 0 (D) 1 (E) 2
2. What is the number of sides of a regular polygon for which the number of diagonals is between 30 and 40 ?
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10
3. If a is a nonzero integer and b is a positive number such that ab2 = log10 b. What is the median of the set {0, 1, a, b,1/b}?
(A) 1/b (B) b (C) a (D) 1 (E) 0
3. Given the function f(x) = 2x4 − 3x3 + 4x2 − 5x + 6, what is the sum of A,B,C,D, and E if f(x) = A(x − 1)4 + B(x − 1)3 + C(x − 1)2 + D(x − 1) + E ?
(A) 16 (B) 17 (C) 18 (D) 19 (E) 20
4. If a polynomial function with real coefficients has 3 distinct x-intercepts, what is the maximal degree of the polynomial?
(A) 3 (B) 4 (C) 5 (D) 6 (E) None of these
5. Let S = i2n−2+i2n+2, where n is an integer and i2 = −1. The total number of possible distinct values of S is
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
6. Given the sequence {an}, where a1 = 1 and an+1 = an + n for n ≥ 1. Find a11.
(A) 50 (B) 55 (C) 60 (D) 65 (E) 66
7. If the age of a person 16 years ago was 5 times the current age of his son, and two years ago the sum of his age and his son’s age was 30, what is the age of his son now?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
8. For a certain integer n, 5n + 16 and 8n + 29 have a common factor larger than one. That common factor is
(A) 11 (B) 13 (C) 17 (D) 19 (E) 23
9. Function f satisfies f(x) + 2f(5 − x) = x for all real numbers x. The value of f(1) is
(A) 7/3 (B) 3/7 (C) 5/2 (D) 2/5 (E) None of these
10. How many 5 digit numbers that are divisible by 3 can be formed using 0,1,2,3,4,5 if no repetitions allowed ?
(A) 216 (B) 120 (C) 240 (D) 126 (E) 96
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