The 77th annual William Lowell Putnam Mathematical Competition took place on December 5th in 705 Thackeray Hall. Eight Pitt students had the mission of solving the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Alex Mang, Andrew Tindall and Jack Hafer. Other participating students were: Matthew Gerstbrein, Terry Tan, Andrew Klang, Tianke Li and Haoming Yan. Below one can find this year’s Putnam problems. Congratulations to all participants!
Problem A1. Find the smallest positive integer such that for every polynomial with integer coefficients and for every integer the integer
(the -th derivative of at ) is divisible by
Problem A2. Given a positive integer , let be the largest integer such that
Problem A3. Suppose that is a function from to such that
for real . (As usual, means and ).
Problem A4. Consider a rectangular region, where and are integers such that The region is to be tiled using tiles of the two types shown:
(The dotted lines divide the tiles into squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.
What is the minimum number of tiles required to tile the region?
Problem A5. Suppose that is a finite group generated by the two elements and , where the order of is odd. Show that every element of can be written in the form
with and . (Here is the number of elements of .)
Problem A6. Find the smallest constant such that every real polynomial of degree that has a root in the interval ,
Problem B1. Let be the sequence such that and for ,
(as usual, the function is the natural logarithm). Show that the infinite series
converges and find its sum.
Problem B2. Define a positive integer to be squarish if either is itself a perfect square of the distance from to the nearest perfect square is a perfect square. For example, is squarish, because the nearest perfect square to is and is a perfect square. (of the positive integers between and , only and are not squarish.) For a positive integer , let be the number of squarish between and inclusive. Find positive constants and such that
or show that no such constants exist.
Problem B3. Suppose that is a finite set of points in the plane such that the area of the triangle is at most whenever and are in . Show that there exists a triangle of area that (together with its interior) covers the set .
Problem B4. Let be a matrix, with entries choasen indepedently at random. Every entry is chosen to be or , each with probability . Find the expected value of (as a function of ), where is the transpose of .
Problem B5. Find all functions from the interval to with the following property:
if and , then .
Problem B6. Evaluate