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

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1. Evaluate
   ∫₀^π/2(cos⁴x + sin x cos³x + sin²x cos²x + sin³x cos x)/(sin⁴x + cos⁴x + 2 sin x cos³x + 2 sin²x cos²x + 2 sin³x cos x) dx.

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) = 3^f(n). What are the last two digits of f (2012)?

5. Determine whether the series ∑_n=2^∞1/ln n - (1/ln n)^(n+1)/n is convergent.

6. Define a sequence (a_n) for n a positive integer inductively by a₁ = 1 and a_n = n/(∏_{d | n}a_d) | 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 A_iA_j = A_k), and tr(A₁ + A₂ + A₃)≠3. Prove that there exists i such that A_iA_j = A_jA_i for all j (here i, j are 1, 2 or 3).