37th Annual
Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 24, 2015
Fill out the individual registration form
 Find all integers n for which
n^{4} +6n^{3} +11n^{2} + 3n + 31 is a
perfect square.
 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 threedimensional region enclosed by the
resulting surface.
 Let
(a_{i})_{1≤i≤2015} be a sequence consisting of 2015
integers, and let
(k_{i})_{1≤i≤2015} be a sequence
of 2015 positive integers (positive integer excludes 0). Let
 A = 
( 
 a_{1}^{k1} 
a_{1}^{k2} 
... 
a_{1}^{k2015} 
 a_{2}^{k1} 
a_{2}^{k2} 
... 
a_{2}^{k2015} 

⋮ 
⋮ 
... 
⋮ 

a_{2015}^{k1} 
a_{2015}^{k2} 
... 
a_{2015}^{k2015} 

). 
Prove that 2015! divides det A.
 Consider the harmonic series
∑_{n≥1}1/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.)
 Evaluate
∫_{0}^{∞}(arctan(πx)  arctan(x))/x dx (where
0≤arctan(x) < π/2 for
0≤x < ∞).
 Let
(a_{1}, b_{1}),...,(a_{n}, b_{n}) be n points in
ℝ^{2} (where
ℝ denotes the real numbers),
and let
ϵ > 0 be a positive
number. Can we find a realvalued function f (x, y) that satisfies
the following three conditions?

f (0, 0) = 1;

f (x, y)≠ 0 for only finitely many
(x, y)∈ℝ^{2};

∑_{r=1}^{r=n} f (x + a_{r}, y + b_{r})  f (x, y) < ϵ for every
(x, y)∈ℝ^{2}.
Justify your answer.
 Let
n be a positive integer and let
x_{1},..., x_{n} be n nonzero
points in
ℝ^{2}. Suppose
⟨x_{i}, x_{j}⟩
(scalar or dot product) is a
rational number for all i, j (
1≤i, j≤n). Let S denote all
points of
ℝ^{2} of the form
∑_{i=1}^{i=n}a_{i}x_{i} where
the a_{i} 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
D_{1}, D_{2},... 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
20151025