36th Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 25, 2014

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1. Find ∑_n=2^n=∞(n² -2n - 4)/(n⁴ +4n² + 16).

2. Evaluate ∫₀²(16 -x²)x/(16 -x² + √((4-x)(4+x)(12+x²))) dx.

3. Find the least positive integer n such that 2²⁰¹⁴ divides 19ⁿ - 1.

4. Suppose we are given a 19×19 chessboard (a table with 19² squares) and remove the central square. Is it possible to tile the remaining 19² - 1 = 360 squares with 4×1 and 1×4 rectangles? (So each of the 360 squares is covered by exactly one rectangle.) Justify your answer.

5. Let n≥1 and r≥2 be positive integers. Prove that there is no integer m such that n(n + 1)(n + 2) = m^r.

6. Let S denote the set of 2 by 2 matrices with integer entries and determinant 1, and let T denote those matrices of S which are congruent to the identity matrix I mod 3 (so

   (

   	a	b
   	c	d

   ) ∈T

   means that a, b, c, d∈ℤ, ad -bc = 1, and 3 divides b, c, a - 1, d - 1; `` ∈" means ``is in").

   (a)

   Let f : T→ℝ (the real numbers) be a function such that for every X, Y∈T with Y≠I, either f (XY) > f (X) or f (XY^-1) > f (X) (or both). Show that given two finite nonempty subsets A, B of T, there are matrices a∈A and b∈B such that if a'∈A, b'∈B and a'b' = ab, then a' = a and b' = b.

   (b)

   Show that there is no f : S→ℝ such that for every X, Y∈S with Y≠±I, either f (XY) > f (X) or f (XY^-1) > f (X).

7. Let A, B be two points in the plane with integer coordinates A = (x₁, y₁) and B = (x₂, y₂). (Thus x_i, y_i∈ℤ, for i = 1, 2.) A path π : A→B is a sequence of down and right steps, where each step has an integer length, and the initial step starts from A, the last step ending at B. In the figure below, we indicated a path from A₁ = (4, 9) to B₁ = (10, 3). The distance d (A, B) between A and B is the number of such paths. For example, the distance between A = (0, 2) and B = (2, 0) equals 6. Consider now two pairs of points in the plane A_i = (x_i, y_i) and B_i = (u_i, z_i) for i = 1, 2, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates):
   • x₂ < x₁ and y₁ > y₂, which means that A₁ is North-East of A₂; u₂ < u₁ and z₁ > z₂, which means that B₁ is North-East of B₂.

   • Each of the points A_i is North-West of the points B_j, for 1≤i, j≤2. In terms of inequalities, this means that x_i < min{u₁, u₂} and y_i > max{z₁, z₂} for i = 1, 2.

   

   (a)

   Find the distance between two points A and B as before, as a function of the coordinates of A and B. Assume that A is North-West of B.

   (b)

   Consider the 2×2 matrix

   M =

   (

   	d (A1, B1)	d (A1, B2)
   	d (A2, B1)	d (A2, B2)

   )

   Prove that for any configuration of points A₁, A₂, B₁, B₂ as described before, det M > 0.