## IMC 2016 – Day 2 – Problem 6

**Problem 6.** Let be a sequence of positive real numbers satisfying . Prove that

## Vojtech Jarnik Competition 2015 – Problems Category 2

**Problem 1.** Let and be two matrices with real entries. Prove that

provided all the inverses appearing on the left-hand side of the equality exist.

**Problem 2.** Determine all pairs of positive integers satisfying the equation

**Problem 3.** Determine the set of real values for which the following series converges, and find its sum:

**Problem 4.** Find all continuously differentiable functions , such that for every the following relation holds:

where

## Divergence

Prove that if then the series diverges.

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## Every expansion has a 0

Suppose is entire such that for each at least one of the coefficients of the expansion is equal to 0. Prove that is a polynomial.

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## Power series & number theory

Let where denotes the number of divisors of . Calculate the radius of convergence of this series and prove that . Furthermore, prove that for .

## Pseudo-alternate series

Suppose is a non-increasing sequence of positive real numbers and such that is convergent.

Prove that .