## Some of the easy Putnam 2016 Problems

Here are a few of the problems of the Putnam 2016 contest. I choose to only list problems which I managed to solve. Most of them are pretty straightforward, so maybe the solutions posted here may be very similar to the official ones. You can find a complete list of the problems on other sites, for example here.

**A1.** Find the smallest integer such that for every polynomial with integer coefficients and every integer , the number

that is the -th derivative of evaluated at , is divisible by .

**Hints.** Successive derivatives give rise to terms containing products of consecutive numbers. The product of consecutive numbers is divisible by . Find the smallest number such that . Prove that does not work by choosing . Prove that works by working only on monomials…

## IMC 2016 – Day 2 – Problem 6

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

## IMC 2016 – Day 1 – Problem 1

**Problem 1.** Let be continuous on and differentiable on . Suppose that has infinitely many zeros, but there is no with .

- (a) Prove that .
- (b) Give an example of such a function.

## SEEMOUS 2016 Problem 4 – Solution

## SEEMOUS 2016 – Problems

**Problem 1.** Let be a continuous and decreasing real valued function defined on . Prove that

When do we have equality?

**Problem 2.** a) Prove that for every matrix there exists a matrix such that .

b) Prove that there exists a matrix such that for all .

**Problem 3.** Let be idempotent matrices () in . Prove that

where and is the set of matrices with real entries.

**Problem 4.** Let be an integer and set

Prove that

a)

b) .

Some hints follow.

## 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

## 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 .