Virginia Tech Regional Mathematics Contest

- Let
*x*_{1}= 1,*x*_{2}= 3, and*x*_{n + 1}= (1/(*n*+ 1))S_{i = 1}^{n}*x*_{i}for*n*= 2, 3,...._{n - > }and give a proof of your answer. - Given that
*a*> 0 and*c*> 0, find a necessary and sufficient condition on*b*so that*ax*^{2}+*bx*+*c*> 0 for all*x*> 0. - Express sinh 3
*x*as a polynomial in sinh*x*. As an example, the identity cos 2*x*= 2cos^{2}*x*- 1 shows that cos 2*x*can be expressed as a polynomial in cos*x*. (Recall that sinh denotes the hyperbolic sine defined by sinh*x*= (*e*^{x}-*e*^{-x})/2.) - Find the quadratic polynomial
*p*(*t*) =*a*_{0}+*a*_{1}*t*+*a*_{2}*t*^{2}such that*t*^{n}*p*(*t*) d*t*=*n*for*n*= 0, 1, 2. - Verify that, for
*f*(*x*) =*x*+ 1,lim_{r - > 0+}((*f*(*x*))^{r}d*x*)^{1/r}=*e*^{ln f(x) dx}. - Sets
*A*and*B*are defined by*A*= {1, 2,...,*n*} and*B*= {1, 2, 3}. Determine the number of distinct functions from*A*onto*B*. (A function*f*:*A*- >*B*is ``onto" if for each*b**B*there exists*a**A*such that*f*(*a*) =*b*.) - A function
*f*from the positive integers to the positive integers has the properties:*f*(1) = 1,*f*(*n*) = 2 if*n*100,-
*f*(*n*) =*f*(*n*/2) if*n*is even and*n*< 100, -
*f*(*n*) =*f*(*n*^{2}+ 7) if*n*is odd and*n*> 1.

- (a)
- Find all positive integers
*n*for which the stated properties require that*f*(*n*) = 1. - (b)
- Find all positive integers
*n*for which the stated properties do not determine*f*(*n*).

- Find all pairs
*N*,*M*of positive integers,*N*<*M*, such thatS_{j=N}^{M}1/(*j*(*j*+ 1)) = 1/10.