Edited by Gerald A. Edgar, Daniel H. Ullman, Douglas B. West with the collaboration of Paul Bracken, Ezra A. Brown, Zachary Franco, Christian Friesen,.
PROBLEMS AND SOLUTIONS Edited by Gerald A. Edgar, Daniel H. Ullman, Douglas B. West with the collaboration of Paul Bracken, Ezra A. Brown, Zachary Franco, Christian Friesen, L´aszl´o Lipt´ak, Rick Luttmann, Frank B. Miles, Lenhard Ng, Leonard Smiley, Kenneth Stolarsky, Richard Stong, Walter Stromquist, Daniel Velleman, and Fuzhen Zhang. Proposed problems should be submitted online at http: // www. americanmathematicalmonthly. submittable. com/ submit. Proposed solutions to the problems below should be submitted by September 30, 2017 via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.
PROBLEMS 11978. Proposed by Hideyuki Ohtsuka, Saitama, Japan. Let Fn be the nth Fibonacci number, with F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 when n ≥ 2. Find ∞ X n=0
(−1)n . cosh Fn cosh Fn+3
11979. Proposed by Zachary Franco, Houston, Texas. Let O and I denote the circumcenter and incenter of a triangle ABC. Are there infinitely many nonsimilar scalene triangles ABC for which the lengths AB, BC, CA, and OI are all integers? 11980. Proposed by George Stoica, Saint John, NB, Canada. Let a1 , . . . , an be a nonincreasing list of positive real numbers, and fix an integer k with 1 ≤ k ≤ n. Prove that there exists a partition {B1 , . . . , Bk } of {1, . . . , n} such that min
1≤ j≤k
X
i∈B j
ai ≥
n X 1 1 ai . min 2 1≤ j≤k k + 1 − j i= j
11981. Proposed by Cezar Lupu, University of Pittsburgh, Pittsburgh, PA. Suppose that f : [0, 1] → R is a differentiable function with continuous derivative and with Z 1 Z 1 f (x) d x = x f (x) d x = 1. 0
0
Prove
Z
1 0
′ 3 f (x) d x ≥
128 3π
2
.
11982. Proposed by Ovidiu Furdui, Mircea Ivan, and Alina Sˆınt˘am˘arian, Technical University of Cluj-Napoca, Cluj-Napoca, Romania. Calculate !1/x ∞ X x n lim . x→∞ n n=1 http://dx.doi.org/10.4169/amer.math.monthly.124.5.465
May 2017]
PROBLEMS AND SOLUTIONS
465
