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 = 90o. 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) = anxn + an-1xn-1 + ... + a1x + a0∈ℝ[x] be a polynomial with real coefficients ai. Suppose that

    an/((n + 1)(n + 2)) + an-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 sS, 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, tS, if st 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.