IMAC 2014

The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$ . Determine $f (2014)$ . $N_0=\{0,1,2,...\}$

A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$ , intersect $XD$ at $Q$ . Let $E$ ( $E\ne A$ ) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$ , and $CQ$ are concurrent.

a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers. b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.