Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 26, 2013
Fill out the individual registration form
I = 3√2∫0x√(1 + cos t)/(17 - 8 cos t)dt. If
0 < x < π and
tan I = 2/√3,
what is x?
- Let ABC be a right-angled triangle with
∠ABC = 90o,
and let D on AB such that AD = 2DB. What is the maximum
possible value of
- Define a sequence (an) for
n≥1 by a1 = 2 and
an+1 = an1+n-3/2. Is (an) convergent
limn→∞an < ∞)?
- A positive integer n is called special if it can be
represented in the form
n = (x2 + y2)/(u2 + v2),
for some positive integers x, y, u, v. Prove that
- 25 is special;
- 2013 is not special;
- 2014 is not special.
- Prove that
x/√(1 + x2) + y/√(1 + y2) + z/√(1 + z2)≤(3√3)/2 for any positive real numbers x, y, z
x + y + z = xyz.
A = Y-1 -X and let B be the inverse of
X-1 + A-1. Find a
matrix M such that
M2 = XY -BY (you may
assume that A and
X-1 + A-1 are invertible).
∑n=1∞n/(2n +2-n)2 + (- 1)nn/(2n -2-n)2.