37th Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 24, 2015

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1. Find all integers n for which n4 +6n3 +11n2 + 3n + 31 is a perfect square.

2. The planar diagram below, with equilateral triangles and regular hexagons, sides length 2cm., is folded along the dashed edges of the polygons, to create a closed surface in three dimensional Euclidean spaces. Edges on the periphery of the planar diagram are identified (or glued) with precisely one other edge on the periphery in a natural way. Thus for example, BA will be joined to QP and AC will be joined to DC. Find the volume of the three-dimensional region enclosed by the resulting surface.

3. Let (ai)1≤i≤2015 be a sequence consisting of 2015 integers, and let (ki)1≤i≤2015 be a sequence of 2015 positive integers (positive integer excludes 0). Let

A = (
 a1k1 a1k2 ... a1k2015 a2k1 a2k2 ... a2k2015 ⋮ ⋮ ... ⋮ a2015k1 a2015k2 ... a2015k2015
).

Prove that 2015! divides det A.

4. Consider the harmonic series n≥11/n = 1 + 1/2 + 1/3.... Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms 1/k.)

5. Evaluate 0(arctan(πx) - arctan(x))/x dx     (where 0≤arctan(x) < π/2 for 0≤x < ∞).

6. Let (a1, b1),...,(an, bn) be n points in 2 (where ℝ denotes the real numbers), and let ϵ > 0 be a positive number. Can we find a real-valued function f (x, y) that satisfies the following three conditions?
1. f (0, 0) = 1;

2. f (x, y)≠ 0 for only finitely many (x, y)∈ℝ2;

3. r=1r=n| f (x + ar, y + br) - f (x, y)| < ϵ for every (x, y)∈ℝ2.

7. Let n be a positive integer and let x1,..., xn be n nonzero points in 2. Suppose xi, xj⟩ (scalar or dot product) is a rational number for all i, j ( 1≤i, jn). Let S denote all points of 2 of the form i=1i=naixi where the ai are integers. A closed disk of radius R and center P is the set of points at distance at most R from P (includes the points distance R from P). Prove that there exists a positive number R and closed disks D1, D2,... of radius R such that
(a)
Each disk contains exactly two points of S;

(b)
Every point of S lies in at least one disk;

(c)
Two distinct disks intersect in at most one point.

Peter Linnell 2015-10-25