33rd Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 29, 2011

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1. Evaluate ∫₁⁴(x - 2)/((x² +4)√x) dx.

2. A sequence (a_n) is defined by a₀ = -1, a₁ = 0, and
   a_n+1 = a_n² - (n + 1)²a_n-1 - 1
   for all positive integers n. Find a₁₀₀.

3. Find ∑_k=1^∞(k² - 2)/((k + 2)!).

4. Let m, n be positive integers and let [a] denote the residue class mod mn of the integer a (thus {[r] | r is an integer} has exactly mn elements). Suppose the set {[ar] | r is an integer} has exactly m elements. Prove that there is a positive integer q such that q is prime to mn and [nq] = [a].

5. Find lim_{x-> ∞}(2x)^1+1/(2x) -x^1+1/x -x.

6. Let S be a set with an asymmetric relation <; this means that if a, b∈S and a < b, then we do not have b < a. Prove that there exists a set T containing S with an asymmetric relation ≺ with the property that if a, b∈S, then a < b if and only if a≺b, and if x, y∈T with x≺y, then there exists t∈T such that x≺t≺y ( t∈T means ``t is an element of T").

7. Let P(x) = x¹⁰⁰ +20x⁹⁹ +198x⁹⁸ + a₉₇x⁹⁷ + ... + a₁x + 1 be a polynomial where the a_i ( 1≤i≤97) are real numbers. Prove that the equation P(x) = 0 has at least one complex root (i.e. a root of the form a + bi with a, b real numbers and b≠ 0).