32nd Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 30, 2010

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1. Let d be a positive integer and let A be a d×d matrix with integer entries. Suppose I + A + A2 + ... + A100 = 0 (where I denotes the identity d×d matrix, so I has 1's on the main diagonal, and 0 denotes the zero matrix, which has all entries 0). Determine the positive integers n≤100 for which An + An+1 + ... + A100 has determinant ±1.

2. For n a positive integer, define f1(n) = n and then for i a positive integer, define fi+1(n) = fi(n)fi(n). Determine f100(75)mod 17 (i.e. determine the remainder after dividing f100(75) by 17, an integer between 0 and 16). Justify your answer.

3. Prove that cos(π/7) is a root of the equation 8x3 -4x2 - 4x + 1 = 0, and find the other two roots.

4. Let ΔABC be a triangle with sides a, b, c and corresponding angles A, B, C (so a = BC and A = ∠BAC etc.). Suppose that 4A + 3C = 540o. Prove that (a - b)2(a + b) = bc2.

5. Let A, B be two circles in the plane with B inside A. Assume that A has radius 3, B has radius 1, P is a point on A, Q is a point on B, and A and B touch so that P and Q are the same point. Suppose that A is kept fixed and B is rolled once round the inside of A so that Q traces out a curve starting and finishing at P. What is the area enclosed by this curve?

6. Define a sequence by a1 = 1, a2 = 1/2, and an+2 = an+1 - anan+1/2 for n a positive integer. Find limn-> ∞nan.

7. Let n=1an be a convergent series of positive terms (so ai > 0 for all i) and set bn = 1/(nan2) for n≥1. Prove that n=1n/(b1 + b2 + ... + bn) is convergent.

Peter Linnell 2010-10-31