## 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 .

## Putnam 2017 A2 – Solution

**Problem A2.** We have the following recurrence relation

for , given and . In order to prove that is always a polynomial with integer coefficients we should prove that divides somehow. Recurrence does not seem to work very well. Also, root based arguments might work, but you need to take good care in the computation.

A simpler idea, which is classic in this context, is to try and linearize the recurrence relation. In order to do this, let’s write two consecutive recurrence relations

We add them and we obtain the following relation

which leads straightforward to a telescoping argument. Finally, we are left with a simple linear recurrence with integer coefficient polynomials, and the result follows immediately.

## 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…

## Putnam 2014 Problems

**A1.** Prove that every nonzero coefficient of the Taylor series of about is a rational number whose numerator (in lowest terms) is either or a prime number.

**A2.** Let be the matrix whose entry in the -th row and -th column is

for Compute

**A3.** Let and for Compute

in closed form.

**A4.** Suppose is a random variable that takes on only nonnegative integer values, with and (Here denotes the expectation of the random variable ) Determine the smallest possible value of the probability of the event

**A5.** Let Prove that the polynomials and are relatively prime for all positive integers and with

**A6.** Let be a positive integer. What is the largest for which there exist matrices and with real entries such that for all and the matrix product has a zero entry somewhere on its diagonal if and only if

**B1.** A *base over-expansion* of a positive integer is an expression of the form with and for all For instance, the integer has two base 10 over-expansions: and the usual base 10 expansion Which positive integers have a unique base 10 over-expansion?

**B2.** Suppose that is a function on the interval such that for all and How large can be?

**B3.** Let be an matrix with rational entries. Suppose that there are at least distinct prime numbers among the absolute values of the entries of Show that the rank of is at least

**B4.** Show that for each positive integer all the roots of the polynomial

are real numbers.

**B5.** In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible matrices with entries in the field of integers modulo where is a fixed positive integer and is a fixed prime number. The rules of the game are:

(1) A player cannot choose an element that has been chosen by either player on any previous turn.

(2) A player can only choose an element that commutes with all previously chosen elements.

(3) A player who cannot choose an element on his/her turn loses the game.

Patniss takes the first turn. Which player has a winning strategy?

**B6.** Let be a function for which there exists a constant such that for all Suppose also that for each rational number there exist integers and such that Prove that there exist finitely many intervals such that is a linear function on each and

## Putnam 2010 A6

Let be a strictly decreasing continuous function such that . Prove that diverges.

*Putnam 2010 A6*

## Putnam 2010 A5

Let be a group, with operation . Suppose that:

- is a subset of ;
- For each either or (or both), where is the usual cross product.

Prove that for all .

*Putnam A5*

## Putnam 2010 A4

Prove that for each positive integer the number is not prime.

*Putnam 2010 A4*