39th Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 21, 2017

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1. Determine the number of real solutions to the equation √(2 -x²) = ³√(3 -x³).

2. Evaluate ∫₀^adx/(1 + cos x + sin x) for - π/2 < a < π. Use your answer to show that ∫₀^π/2dx/(1 + cos x + sin x) = ln 2.

3. Let ABC be a triangle and let P be a point in its interior. Suppose ∠BAP = 10^o, ∠ABP = 20^o, ∠PCA = 30^o and ∠PAC = 40^o. Find ∠PBC.

4. Let P be an interior point of a triangle of area T. Through the point P, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let a, b and c be the areas of the three triangles. Prove that √T = √a + √b + √c.

5. Let f (x, y) = (x + y)/2, g(x, y) = √xy, h(x, y) = 2xy/(x + y), and let
   S = {(a, b)∈ℕ×ℕ | a≠b andf (a, b), g(a, b), h(a, b)∈ℕ},
   where ℕ denotes the positive integers. Find the minimum of f over S.

6. Let f (x)∈ℤ[x] be a polynomial with integer coefficients such that f (1) = -1, f (4) = 2 and f (8) = 34. Suppose n∈ℤ is an integer such that f (n) = n² - 4n - 18. Determine all possible values for n.

7. Find all pairs (m, n) of nonnegative integers for which
   m² +2·3ⁿ = m(2ⁿ⁺¹ - 1).