## Vojtech Jarnik Competition 2015 – Category 1 Problems

**Problem 1.** Let be differentiable on . Prove that there exists such that

**Problem 2.** Consider the infinite chessboard whose rows and columns are indexed by positive integers. Is it possible to put a single positive rational number into each cell of the chessboard so that each positive rational number appears exactly once and the sum of every row and of every column is finite?

**Problem 3.** Let and . Determine for each of the polynomials and whether it is a divisor of some nonzero polynomial whose coefficients are all in the set .

**Problem 4.** Let be a positive integer and let be a prime divisor of . Suppose that the complex polynomial with and is divisible by the cyclotomic polynomial . Prove that there at least non-zero coefficients .

The cyclotomic polynomial is the monic polynomial whose roots are the -th primitive complex roots of unity. Euler’s totient function denotes the number of positive integers less than or eual to which are coprime to .