The 76th annual William Lowell Putnam Mathematical Competition took place today, December 5th in 703 Thackeray Hall. Eleven Pitt students had the mission of tearing down the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Stefan Ivanovici, Matthew Smylie and Ming Tian. Other participating students were: Alec Jasen, Derek Orr, Tommy Bednar, Alex Mang, Jacob Gross, Andrew Tindall, Jack Hafer, and Mark Paulson. Below one can find this year’s Putnam problems. Congratulations to all participants!
Problem A1. Let and be points on the same branch of the hyperbola . Suppose that is a point lying between and on this hyperbola, such that the area of the triangle is as large as possible. Show that the region bounded by the hyperbola and the chord has the same area as the region bounded by the hyperbola and the chord .
Problem A2. Let and , and for . Find an odd prime factor of .
Problem A3. Compute
Problem A4. For each number , let
where is the set of positive integers for which is even. What is the largest real number such that for all ?
(As usual, denotes the greatest integer less or equal to .)
Problem A5. Let an odd positive integer, and let denote the number of integers such that and . Show that is odd if and only if is of the form with a positive integer and a prime congruent to or modulo .
Problem A6. Let be a positive integer. Suppose that are matrices with real entries such that , and such that and have the same characteristic polynomial. Prove that for every matrix with real entries.
Problem B1. Let be a three times differentiable function (defined on and real-valued) such that has at least five distinct real zeros. Prove that has at least two distinct real zeros.
Problem B2. Given a list of the positive integers , take the first three numbers and their sum and cross all four numbers off the list. Repeat with the three smallest remaining numbers and their sum . Continue this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: . Prove or disprove that there is some number in this sequence whose base representation ends with .
Problem B3. Let be the set of real matrices
whose entries (in that order) form an arithmetic progression. Find all matrices in for which there is some integer such that is also in .
Problem B4. Let be the set of all triples of positive integers for which there exist triangles with side lengths . Express
as a rational number in lowest terms.
Problem B5. Let be the number of permutations of such that
for all . Show that for , the quantity
does not depend on , and find its value.
Problem B6. For each positive integer , let be the number of odd divisors of in the interval . Evaluate