## 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…

**A2.** Given a positive integer let be the largest integer such that

Evaluate

**Hints:** Write formulas for binomial coefficients and simplify what is possible. What remains should be an inequation of degree in terms of . Find the largest positive root of the equation and the largest should be the integer part of that root. Using the fact that the integer part of is equivalent to as we can conclude.

**A3.** Suppose that is a function from such that

for all real . Find .

**Hints:** Use a classical trick of replacing by . This makes the second argument equal to . Note that making now the second argument becomes . This will give three equations which allow us to find a formula for .

**B1.** Let be a sequence such that and for

Show that the infinite series converges and find its sum.

**Hints:** First note that and see that . Exponentiate to find which will give the monotonicity of the sequence (and the convergence) and also a telescopic expression for . Summing becomes a simple expresion in terms of the limit of .

**B3.** Suppose that is a finite set of points in the plane such that the area of the triangle is at most whenever are in . Show that there exists a triangle of area that together with its interior covers the set .

**Hints:** I was surprised to see this, since it is a classical problem… Just pick the triangle with vertices in which has the largest area… Then consider the triangle whose midpoints are . This triangle will contain .

One way to make this more challenging would be to consider a bounded set in the plane, not necessarily finite… I’ll come back in the next days with more solutions. Problem A4 was particularily interesting and I’ll dedicate a post in order to present it.