## Putnam 2017 A3 – Solution

**Problem A3.** Denote . Then is continuous and . We can see that

Now note that on we have so . Furthermore, on we have so multiplying with we get . Therefore

To prove that goes to we can still work with . Note that the negative part cannot get too big:

As for the positive part, taking we have

Next, note that on

We would need that the last term be larger than . This is equivalent to . Since is continuous on , it is bounded above, so some delta small enough exists in order for this to work.

Grouping all of the above we get that

Since we get that goes to .

## 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