40th Annual Virginia Tech Regional Mathematics Contest
From 9:00 a.m. to 11:30 a.m., October 27, 2018

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1. It is known that ∫₁²x^-1arctan(1 + x) dx = qπln(2) for some rational number q. Determine q. Here 0≤arctan(x) < π/2 for 0≤x < ∞.

2. Let A, B∈M₆(ℤ) such that A≡I≡B mod 3 and A³B³A³ = B³. Prove that A = I. Here M₆(ℤ) indicates the 6 by 6 matrices with integer entries, I is the identity matrix, and X≡Y mod 3 means all entries of X -Y are divisible by 3.

3. Prove that there is no function f : ℕ→ℕ such that f (f (n)) = n + 1. Here ℕ is the positive integers {1, 2, 3,...}.

4. Let m, n be integers such that n≥m≥1. Prove that

   n^-1gcd(m, n)

   (

   	n
   	m

   )

   is an integer. Here gcd denotes greatest common divisor and

   (

   	n
   	m

   )

   = n!/(m!(n -m)!) denotes the binomial coefficient.

5. For n∈ℕ, let a_n = ∫₀^1/√n| 1 + e^it + e^2it + ... + e^nit| dt. Determine whether the sequence (a_n) = a₁, a₂,... is bounded.

6. For n∈ℕ, define a_n = (1 + 1/3 + 1/5 + ... + 1/(2n - 1))/(n + 1) and
   b_n = (1/2 + 1/4 + 1/6 + ... + 1/(2n))/n. Find the maximum and minimum of a_n -b_n for 1≤n≤999.

7. A continuous function f : [a, b]→[a, b] is called piecewise monotone if [a, b] can be subdivided into finitely many subintervals
   I₁ = [c₀, c₁], I₂ = [c₁, c₂], ..., I_ℓ = [c_ℓ-1, c_ℓ]
   such that f restricted to each interval I_j is strictly monotone, either increasing or decreasing. Here we are assuming that a = c₀ < c₁ < ... < c_ℓ-1 < c_ℓ = b. We are also assuming that each I_j is a maximal interval on which f is strictly monotone. Such a maximal interval is called a lap of the function f, and the number ℓ = ℓ(f ) of distinct laps is called the lap number of f. If f : [a, b]→[a, b] is a continuous piecewise-monotone function, show that the sequence ( ⁿ√ℓ(fⁿ)) converges; here fⁿ means f composed with itself n-times, so f²(x) = f (f (x)) etc.