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dicular through D to AB to obtain the orthogonal projection I of D onto AB. ... Note that A and B as well as H and I are symmetric about the midpoint M of AB.
PROBLEMS AND SOLUTIONS EDITORS

Curtis Cooper CMJ Problems University of Central Missouri

Jerzy Wojdylo

Charles N. Curtis

CMJ Problems, Elect Department of Mathematics Southeast Missouri State University One University Plaza Cape Girardeau, MO 63701 [email protected]

CMJ Solutions Mathematics Department Missouri Southern State University 3950 Newman Road Joplin, MO 64801 [email protected]

This section contains problems intended to challenge students and teachers of college mathematics. We urge you to participate actively both by submitting solutions and by proposing problems that are new and interesting. To promote variety, the editors welcome problem proposals that span the entire undergraduate curriculum. Proposed problems should be sent to Jerzy Wojdylo, either by email (preferred) as a pdf, TEX, or Word attachment or by mail to the address provided above. Whenever possible, a proposed problem should be accompanied by a solution, appropriate references, and any other material that would be helpful to the editors. Proposers should submit problems only if the proposed problem is not under consideration by another journal. Solutions to the problems in this issue should be sent to Chip Curtis, either by email as a pdf, TEX, or Word attachment (preferred) or by mail to the address provided above, no later than June 15, 2017.

PROBLEMS 1111. Proposed by Greg Oman, University of Colorado, Colorado Springs, CO. P Let ∞ n=0 x n be a real convergent Pseries with positive terms. Prove that there is a subsequence {xnk } of {xn } such that ∞ k=0 x n k is irrational.

1112. Proposed by Ovidiu Furdui, Technical University of Cluj-Napoca, Cluj-Napoca, Romania. Prove the following statements are equivalent for a real 2 × 2 matrix A. (a) cosh A is singular. (b) cosh A = 02 (the 2 × 2 zero matrix).   (2m−1)π 0 2  P −1 for some integer m and some real invertible (c) A = P  −(2m−1)π 0 2 matrix P. http://dx.doi.org/10.4169/college.math.j.48.5.370

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 THE MATHEMATICAL ASSOCIATION OF AMERICA

1113. Proposed by Ovidiu Furdui, Technical University of Cluj-Napoca, Cluj-Napoca, Romania. Let a and b 6= 0 be real numbers. Calculate  n b 1 − na2 n  . lim  b n→∞ −n 1 + na2 1114. Proposed by Ovidiu Furdui and Alina Sˆınt˘am˘arian, Technical University of ClujNapoca, Cluj-Napoca, Romania. Find all differentiable functions f : R → R with f (0) = 1 such that, for all x ∈ R, f ′ (x) = x 2 f 2 (−x) f (x). 1115. Proposed by Mehtaab Sawhney (student), Massachusetts Institute of Technology, Cambridge, MA. Z 1 x Z 1 e −1 2 ex d x > d x. (a) Prove that x 0 0 Z 1 Z 1Z 1Z 1 Z 1 x2 e −1 3 (b) Prove that ex d x + e x yz d x d y dz > 2 d x. x2 0 0 0 0 0

SOLUTIONS A triangle construction 1086. Proposed by Michel Bataille, Rouen, France. Let AB be a given line segment with length c > 0. Construct a right-angled triangle whose legs a, b satisfy a 2/3 + b2/3 = c2/3 using a ruler and square set (a drafting tool with a right angle, also known as a set square). Solution by the proposer. We can readily construct C and D such that ABCD is a rectangle. Draw the perpendicular through D to AB to obtain the orthogonal projection I of D onto AB. Similarly, J and K are the orthogonal projections of I onto AC and BC, respectively. We prove that the right-angled triangle JIK answers the question, that is, if IJ = a and IK = b, then a 2/3 + b2/3 = c2/3 . Let H be the orthogonal projection of C onto AB. Note that A and B as well as H and I are symmetric about the midpoint M of AB. AI IJ BI IK We have AB = BC and AB = AC , hence IJ · IK = =

AI · BI · AC · BC HA · HB · AC · BC = 2 AB AB2 HC2 · HC · AB HC3 = AB AB2

IJ since HA · HB = HC2 and AC · BC = HC · AB. We also have IK = AI·BC , and AI · BI·AC 2 2 IJ = AB = BH · BA = BC , and BI · BA = AH · AB = AC . It readily follows that IK 3 BC . Thus, AC3

VOL. 48, NO. 5, NOVEMBER 2017 THE COLLEGE MATHEMATICS JOURNAL

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