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=2n=∞(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) = mr.

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, YT with YI, 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 aA and bB 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, YS 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 xi, yi∈ℤ, for i = 1, 2.) A path π : AB 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 Ai = (xi, yi) and Bi = (ui, zi) 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 Ai is North-West of the points Bj, for 1≤i, j≤2. In terms of inequalities, this means that xi < min{u1, u2} and yi > 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.

Peter Linnell 2014-10-24