34th Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 27, 2012

Fill out the individual registration form

1. Evaluate

0π/2(cos4x + sin x cos3x + sin2x cos2x + sin3x cos x)/(sin4x + cos4x + 2 sin x cos3x + 2 sin2x cos2x + 2 sin3x cos xdx.

2. Solve in real numbers the equation 3x -x3 = √(x + 2).

3. Find nonzero complex numbers a, b, c, d, e such that

 a + b + c + d + e = -1 a2 + b2 + c2 + d2 + e2 = 15 1/a + 1/b + 1/c + 1/d + 1/e = -1 1/a2 +1/b2 +1/c2 +1/d2 +1/e2 = 15 abcde = -1

4. Define f (n) for n a positive integer by f (1) = 3 and f (n + 1) = 3f(n). What are the last two digits of f (2012)?

5. Determine whether the series n=21/ln n - (1/ln n)(n+1)/n is convergent.

6. Define a sequence (an) for n a positive integer inductively by a1 = 1 and an = n/(∏d | nad) | 1≤d < n. Thus a2 = 2, a3 = 3, a4 = 2 etc. Find a999000.

7. Let A1, A2, A3 be three pairwise unequal 2×2 matrices with entries in ℂ (the complex numbers). Let tr denote the trace of a matrix (so
tr( (
 a b c d
) ) = a + d)

Suppose {A1, A2, A3} is closed under matrix multiplication (i.e. given i, j, there exists k such that AiAj = Ak), and tr(A1 + A2 + A3)≠3. Prove that there exists i such that AiAj = AjAi for all j (here i, j are 1, 2 or 3).

Peter Linnell 2012-10-28