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 + x²) dx.

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 M_n(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 M_n(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 f_k(X) in the variable X, depending on q and k, by f_k(X) = ∏_i=0^k-1(1 -qⁱX)(1 -qⁱ⁺¹X^-1). (Here ∏ denotes product.) Show that the constant term of f_k(X), i.e. the coefficient of X⁰ in f_k(X), is equal to
   (1 -q^k+1)(1 -q^k+2)...(1 -q^2k)/((1 -q)(1 -q²)...(1 -q^k)).