Virginia Tech Regional Mathematics Contest

- A circle
*C*of radius*r*is circumscribed by a parallelogram*S*. Let q denote one of the interior angles of*S*, with 0 < qp/2. Calculate the area of*S*as a function of*r*and q. - A man goes into a bank to cash a check. The teller mistakenly
reverses the amounts and gives the man cents for dollars and dollars
for cents. (Example: if the check was for $5.10, the man was given
$10.05.). After spending five cents, the man finds that he still
has twice as much as the original check amount. What was the
original check amount? Find all possible solutions.
- Find the general solution of
*y*(*x*) +*y*(*t*) d*t*=*x*^{2}. - Let
*a*be a positive integer. Find all positive integers*n*such that*b*=*a*^{n}satisfies the condition that*a*^{2}+*b*^{2}is divisible by*ab*+ 1. - Let
*f*be differentiable on [0, 1] and let*f*(a) = 0 and*f*(*x*_{0}) = - .0001 for some a and*x*_{0}(0, 1). Also let |*f'*(*x*)|2 on [0, 1]. Find the smallest upper bound on |a -*x*_{0}| for all such functions. - Find positive real numbers
*a*and*b*such that*f*(*x*) =*ax*-*bx*^{3}has four extrema on [- 1, 1], at each of which |*f*(*x*)| = 1. - For any set
*S*of real numbers define a new set*f*(*S*) by*f*(*S*) = {*x*/3 |*x**S*} {(*x*+ 2)/3 |*x**S*}.- (a)
- Sketch, carefully, the set
*f*(*f*(*f*(*I*))), where*I*is the interval [0, 1]. - (b)
- If
*T*is a bounded set such that*f*(*T*) =*T*, determine,*with proof*, whether*T*can contain 1/2.

- Let
*T*(*n*) be the number of incongruent triangles with integral sides and perimeter*n*6. Prove that*T*(*n*) =*T*(*n*- 3) if*n*is even, or disprove by a counterexample. (*Note*: two triangles are*congruent*if there is a one-to-one correspondence between the sides of the two triangles such that corresponding sides have the same length.)