PROBLEMS AND SOLUTIONS Edited by Gerald A. Edgar, Doug Hensley, Douglas B. West with the collaboration of Mario Benedicty, Itshak Borosh, Paul Bracken, Ezra A. Brown, Randall Dougherty, Tam´ as Erd´ elyi, Zachary Franco, Christian Friesen, Ira M. Gessel, L´ aszl´ o Lipt´ ak, Frederick W. Luttmann, Vania Mascioni, Frank B. Miles, Richard Pfiefer, Cecil C. Rousseau, Leonard Smiley, Kenneth Stolarsky, Richard Stong, Walter Stromquist, Daniel Ullman, Charles Vanden Eynden, Sam Vandervelde, and Fuzhen Zhang. Proposed problems and solutions should be sent in duplicate to the MONTHLY problems address on the inside front cover. Submitted solutions should arrive at that address before May 31, 2009. Additional information, such as generaliza- tions and references, is welcome. The problem number and the solver’s name and address should appear on each solution. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available. PROBLEMS 11404. Proposed by Raimond Struble, North Carolina State at Raleigh, Raleigh, NC. Any three non-concurrent cevians of a triangle create a subtriangle. Identify the sets of non-concurrent cevians which create a subtriangle whose incenter coincides with the incenter of the primary triangle. (A cevian of a triangle is a line segment joining a vertex to an interior point of the opposite edge.) 11405. Proposed by Ovidiu Furdui, Campia Turzii, Cluj, Romania. Let P be an interior point of a tetrahedron ABCD. When X is a vertex, let X ′ be the intersection of the opposite face with the line through X and P . Let XP denote the length of the line segment from X to P . (a) Show that PA · PB · PC · PD ≥ 81 PA ′ · PB ′ · PC ′ · PD ′ , with equality if and only if P is the centroid of ABCD. (b) When X is a vertex, let X ′′ be the foot of the perpendicular from P to the plane of the face opposite X . Show that PA · PB · PC · PD = 81 PA ′′ · PB ′′ · PC ′′ · PD ′′ if and only if the tetrahedron is regular and P is its centroid. 11406. Proposed by A. A. Dzhumadil’daeva, Almaty, Republics Physics and Mathe- matics School, Almaty, Kazakhstan. Let n!! denote the product of all positive integers not greater than n and congruent to n mod 2, and let 0!! = (−1)!! = 1. Thus, 7!! = 105 and 8!! = 384. For positive integer n, find n  i =0  n i  (2i − 1)!! (2(n − i ) − 1)!! in closed form. 11407. Proposed by Erwin Just (Emeritus), Bronx Community College of the City Uni- versity of New York, New York, NY. Let p be prime greater than 3. Does there exists a ring with more than one element (not necessarily having a multiplicative identity) such that for all x in the ring, ∑ p i =1 x 2i −1 = 0? 82 c  THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 116