Solve the equation
- The function has derivatives. Show that if and then we can find such that .
For , let where the sum ranges over all pairs of positive integer satisfying the indicated inequalities. Evaluate:
- This problem deals to elementary functional analysis and is taken from very old paper of Putnam Competitions.
has continuous second derivative and , for all .
are integers such that are all integers. Put and .
• Show that for , there are atmost such indices .
• Show that there are atmost lattice points on the curve .