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)√xdx.

2. A sequence (an) is defined by a0 = -1, a1 = 0, and

an+1 = an2 - (n + 1)2an-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 limx-> ∞(2x)1+1/(2x) -x1+1/x -x.

6. Let S be a set with an asymmetric relation <; this means that if a, bS 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, bS, then a < b if and only if ab, and if x, yT with xy, then there exists tT such that xty ( tT means t is an element of T").

7. Let P(x) = x100 +20x99 +198x98 + a97x97 + ... + a1x + 1 be a polynomial where the ai ( 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).

Peter Linnell 2011-10-31