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William Lowell Putnam Mathematical Competition at the University of Pittsburgh, 2015

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!

SESSION A:

Problem A1. Let A and B be points on the same branch of the hyperbola xy=1. Suppose that P is a point lying between A and B on this hyperbola, such that the area of the triangle APB is as large as possible. Show that the region bounded by the hyperbola and the chord AP has the same area as the region bounded by the hyperbola and the chord PB.

Problem A2. Let a_{0}=1 and a_{1}=2, and a_{n}=4a_{n-1}-a_{n-2} for n\geq 2. Find an odd prime factor of a_{2015}.

Problem A3. Compute

\displaystyle\log_{2}\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right).

Problem A4. For each number x, let

\displaystyle f(x)=\sum_{n\in S_{x}}\frac{1}{2^n},

where S_{x} is the set of positive integers n for which [nx] is even. What is the largest real number L such that f(x)\geq L for all x\in (0,1)?

(As usual, [z] denotes the greatest integer less or equal to z.)

Problem A5. Let q an odd positive integer, and let N_{q} denote the number of integers a such that 0<a<\frac{q}{4} and gdc(a, q)=1. Show that N_{q} is odd if and only if q is of the form p^{k} with k a positive integer and p a prime congruent to 5 or 7 modulo 8.

Problem A6. Let n be a positive integer. Suppose that A, B are n\times n matrices with real entries such that AM=MB, and such that A and B have the same characteristic polynomial. Prove that \displaystyle\det (A-MX)=\det (B-XM) for every n\times n matrix X with real entries.

 

SESSION B:

 

Problem B1. Let f be a three times differentiable function (defined on \mathbb{R} and real-valued) such that f has at least five distinct real zeros. Prove that \displaystyle f+6f^{\prime}+12f^{{\prime}{\prime}}+8f^{{\prime}{\prime}{\prime}} has at least two distinct real zeros.

Problem B2. Given a list of the positive integers 1, 2, 4, \ldots, take the first three numbers 1,2, 3 and their sum 6 and cross all four numbers off the list. Repeat with the three smallest remaining numbers 4, 5, 7 and their sum 16. Continue this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: 6, 16, 27, 36, \ldots. Prove or disprove that there is some number in this sequence whose base 10 representation ends with 2015.

Problem B3. Let S be the set of 2\times 2 real matrices

\displaystyle M= \begin{pmatrix} a & b \\ c & d \end{pmatrix}

whose entries a, b, c, d (in that order) form an arithmetic progression. Find all matrices M in S for which there is some integer k>1 such that M^k is also in S.

Problem B4. Let T be the set of all triples (a, b, c) of positive integers for which there exist triangles with side lengths a, b, c. Express

\displaystyle\sum_{(a, b, c)\in T}\frac{2^a}{3^b5^c}

as a rational number in lowest terms.

Problem B5. Let P_{n} be the number of permutations \pi of \{1, 2, \ldots, n\} such that

|i-j|=1 implies |\pi (i)-\pi (j)|\leq 2

for all i, j\in \{1, 2, \ldots, n\}. Show that for n\geq 2, the quantity

\displaystyle P_{n+5}-P_{n+4}-P_{n+3}+P_{n}

does not depend on n, and find its value.

Problem B6. For each positive integer k, let A(k) be the number of odd divisors of k in the interval [1, \sqrt{2k}). Evaluate

\displaystyle\sum_{k=1}^{\infty} (-1)^{k-1}\frac{A(k)}{k}.


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