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
n4 +6n3 +11n2 + 3n + 31 is a
- 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
(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 =
Prove that 2015! divides det A.
- 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.)
∫0∞(arctan(πx) - arctan(x))/x dx (where
0≤arctan(x) < π/2 for
0≤x < ∞).
(a1, b1),...,(an, bn) be n points in
ℝ denotes the real numbers),
ϵ > 0 be a positive
number. Can we find a real-valued function f (x, y) that satisfies
the following three conditions?
Justify your answer.
f (0, 0) = 1;
f (x, y)≠ 0 for only finitely many
∑r=1r=n| f (x + ar, y + br) - f (x, y)| < ϵ for every
n be a positive integer and let
x1,..., xn be n nonzero
(scalar or dot product) is a
rational number for all i, j (
1≤i, j≤n). Let S denote all
ℝ2 of the form
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
- Each disk contains exactly two points of S;
- Every point of S lies in at least one disk;
- Two distinct disks intersect in at most one point.