Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 22, 2016
Fill out the individual registration form
∫12(ln x)/(2 - 2x + x2) dx.
- Determine the real numbers k such that
- Let n be a positive integer and let
the n by n matrices with entries from the integers mod 2. If
that the number of matrices A in
A2 = 0 (the matrix with all entries zero) is an even positive
- 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).
- 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.
A, B, P, Q, X, Y be square matrices of the same size. Suppose that
|A + B + AB
||AX = XQ
|P + Q + PQ
||PY = YB.
Prove that AB = BA.
- 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).
∏ 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)).