35th Annual Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 26, 2013

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1. Let I = 3√2∫0x√(1 + cos t)/(17 - 8 cos t)dt. If 0 < x < π and tan I = 2/√3, what is x?

2. 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 ACD?

3. Define a sequence (an) for n≥1 by a1 = 2 and an+1 = an1+n-3/2. Is (an) convergent (i.e. limn→∞an < ∞)?

4. 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
(a)
25 is special;

(b)
2013 is not special;

(c)
2014 is not special.

5. Prove that x/√(1 + x2) + y/√(1 + y2) + z/√(1 + z2)≤(3√3)/2 for any positive real numbers x, y, z such that x + y + z = xyz.

6. Let
X = (
 7 8 9 8 -9 -7 -7 -7 9
)

Y = (
 9 8 -9 8 -7 7 7 9 8
)

let 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).

7. Find n=1n/(2n +2-n)2 + (- 1)nn/(2n -2-n)2.

Peter Linnell 2013-11-06