38th Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 22, 2016

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1. Evaluate 12(ln x)/(2 - 2x + x2dx.

2. Determine the real numbers k such that n=1((2n)!/(4nn!n!))k is convergent.

3. Let n be a positive integer and let Mn(ℤ2) denote the n by n matrices with entries from the integers mod 2. If n≥2, prove that the number of matrices A in Mn(ℤ2) satisfying A2 = 0 (the matrix with all entries zero) is an even positive integer.

4. For a positive integer a, let P(a) denote the largest prime divisor of a2 + 1. Prove that there exist infinitely many triples (a, b, c) of distinct positive integers such that P(a) = P(b) = P(c).

5. Suppose that m, n, r are positive integers such that

1 + m + n√3 = (2 + √3)2r-1.

Prove that m is a perfect square.

6. Let A, B, P, Q, X, Y be square matrices of the same size. Suppose that

 A + B + AB = XY AX = XQ P + Q + PQ = YX PY = YB.

Prove that AB = BA.

7. Let q be a real number with | q|≠1 and let k be a positive integer. Define a Laurent polynomial fk(X) in the variable X, depending on q and k, by fk(X) = ∏i=0k-1(1 -qiX)(1 -qi+1X-1). (Here ∏ denotes product.) Show that the constant term of fk(X), i.e. the coefficient of X0 in fk(X), is equal to

(1 -qk+1)(1 -qk+2)...(1 -q2k)/((1 -q)(1 -q2)...(1 -qk)).

Peter Linnell 2016-10-22