Putnam and Beyond

102.49 A very short proof of Pamfilos's characterisation of the rhombus

“…These have appeared in [4] as Problems 223 and 224, respectively, with solutions on pages 297-298; and (2) has also appeared in [3] as Problem 649 on page 230 with a solution on pages 648-649. It is interesting that (1) has also appeared in [5] as Problem 7 in the list of 12 open problems posed on pages 132-133.…”

“…We were motivated to write it up by the recent appearance of two Notes in this Gazette that have the same theme, namely [1] and [2]. The context that this Note describes is concerned with what we call long altitudes, a side issue that came up several years ago when the two firstnamed authors were investigating what they called equicevian points of a triangle; see [3].…”

More Characterizations of Certain Special Families of Simplices

“…Orthocentric tetrahedra are also characterized by the property that the perpendiculars of the faces at their centroids are concurrent. This appears in [13], and also in [6, 1 • , §68, p. 135] and in [12]. In [6, §74.b, p. 154], the equivalence of (4), (5), and (6) is established in 1 • and 2 • , and it is proved in 4 • that a tetrahedron is orthocentric if and only if the three segments joining the midpoints of the three pairs of disjoint edges are equal in length.…”

“…This is reproduced in [12,Problem 586,p. 206], and it also appears in [25] and in [6, 1 • , §68, p. 135].…”

Four quotient set gems

“…Proof. We require the Stolz-Cesàro Theorem [6], which tells us that if x n and y n are two increasing and unbounded sequences of real numbers, then Arithmetic progressions Our final gem concerns arithmetic progressions of natural numbers. Recall that an arithmetic progression with common difference b and length n is a sequence of the form a, a + b, a + 2b, .…”