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

Fill out the individual registration form

1. Let I = 3√2∫₀^x√(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 = 90^o, and let D on AB such that AD = 2DB. What is the maximum possible value of ∠ACD?

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

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=1^∞n/(2ⁿ +2^-n)² + (- 1)ⁿn/(2ⁿ -2^-n)².