41st Annual Virginia Tech Regional Mathematics Contest
From 9:00 a.m. to 11:30 a.m., October 26, 2019

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1. For each positive integer n, define f (n) to be the sum of the digits of 2771ⁿ (so f (1) = 17). Find the minimum value of f (n) (where n≥1). Justify your answer.

2. Let X be the point on the side AB of the triangle ABC such that BX/XA = 9. Let M be the midpoint of BX and let Y be the point on BC such that ∠BMY = 90^o. Suppose AC has length 20 and that the area of the triangle XYC is 9/100 of the area of the triangle ABC. Find the length of BC.

3. Let n be a nonnegative integer and let f (x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀∈ℝ[x] be a polynomial with real coefficients a_i. Suppose that
   a_n/((n + 1)(n + 2)) + a_n-1/(n(n + 1)) + ... + a₁/6 + a₀/2 = 0.
   Prove that f (x) has a real zero.

4. Compute ∫₀¹x²/(x + √(1-x²)) dx (the answer is a rational number).

5. Find the general solution of the differential equation
   x⁴d²y/dx² +2x²dy/dx + (1 - 2x)y = 0
   valid for 0 < x < ∞.

6. Let S be a subset of ℝ with the property that for every s∈S, there exists ϵ > 0 such that (s - ϵ, s + ϵ)∩S = {s}. Prove there exists a function f : S→ℕ, the positive integers, such that for all s, t∈S, if s≠t then f (s)≠f (t).

7. Let S denote the positive integers that have no 0 in their decimal expansion. Determine whether ∑_n∈Sn^-99/100 is convergent.