Virginia Tech Regional Mathematics Contest

- Evaluate
*xe*^{(x4 + 2x2y2 + y4)}d*xdy*. - For each rational number
*r*, define*f*(*r*) to be the smallest positive integer*n*such that*r*=*m*/*n*for some integer*m*, and denote by*P*(*r*) the point in the (*x*,*y*) plane with coordinates*P*(*r*) = (*r*, 1/*f*(*r*)). Find a necessary and sufficient condition that, given two rational numbers*r*_{1}and*r*_{2}such that 0 <*r*_{1}<*r*_{2}< 1,*P*((*r*_{1}*f*(*r*_{1}) +*r*_{2}*f*(*r*_{2}))/(*f*(*r*_{1}) +*f*(*r*_{2})))*r*_{1}, 0) and*P*(*r*_{2}) with the line joining*P*(*r*_{1}) and (*r*_{2}, 0). - Solve the differential equation
*y*^{y}=*e*^{dy/dx}with the initial condition*y*=*e*when*x*= 1. - Let
*f*(*x*) be a twice continuously differentiable in the interval (0,). Iflimfind lim_{x - > }(*x*^{2}*f''*(*x*) + 4*xf'*(*x*) + 2*f*(*x*)) = 1,_{x - > }*f*(*x*) and lim_{x - > }*xf'*(*x*). Do**not**assume any special form of*f*(*x*). Hint: use l'Hôpital's rule. - Let
*a*_{i},*i*= 1, 2, 3, 4, be real numbers such that*a*_{1}+*a*_{2}+*a*_{3}+*a*_{4}= 0. Show that for arbitrary real numbers*b*_{i},*i*= 1, 2, 3, the equation*a*_{1}+*b*_{1}*x*+ 3*a*_{2}*x*^{2}+*b*_{2}*x*^{3}+ 5*a*_{3}*x*^{4}+*b*_{3}*x*^{5}+ 7*a*_{4}*x*^{6}= 0*x*1. - There are 2
*n*balls in the plane such that no three balls are on the same line and such that no two balls touch each other.*n*balls are red and the other*n*balls are green. Show that there is at least one way to draw*n*line segments by connecting each ball to a unique different colored ball so that no two line segments intersect. - Let us define
*f*_{n, 0}(*x*)= *x*+ ()/*n*for *x*> 0,*n*1,*f*_{n, j + 1}(*x*)= *f*_{n, 0}(*f*_{n, j}(*x*)),*j*= 0, 1,...,*n*- 1.

Find lim_{n - > }*f*_{n, n}(*x*) for*x*> 0.