The American Mathematical Monthly (AMM) has one of the widest readership and highest ranking in the mathematical community. It publishes Expository and Research papers as well as 2-4 Mathematical Notes each issue, ten issues each year. The range of topics is very varied from elementary to highly advanced mathematics. It also publishes a very prestigious and interesting section of proposed problems, currently 6 each issue. Below there is a selection of such proposed problems which I published in the AMM.

List of the published AMM proposals

1. Problem 10970, 109 AMM (2002), p. 854

Let ABC be an acute triangle and let P be a point in its interior. Denote by a, b, c the lengths of the triangle’s sides, by the distances from P to the triangles’ sides and by the distances from P to the vertices A, B, C, respectively. Show that the following double inequality is true

.

When is the equality possible?

2. Problem 10955, 109 AMM (2002), p. 570

Let ABC be a triangle and denote by O, I, H, and G the circumcenter, incenter, orthocenter and centroid of ABC respectively. Let r be the radius of the inscribed circle and R the radius of the circumscribed circle. Show that

from which it follows that triangle OGI is obtuse. Further establish the following inequalities:

, and

and show that in each case the respective constants are the best possible.

3. Problem 10880, 108 AMM (2001), p. 565

Let R and r be the circumradius and inradius respectively of triangle ABC. 

a) Show that ABC has a median whose length is at most 2R - r.

b) Show that ABC has an altitude whose length is at least 2R - r.

4. Problem 10814, 107 AMM (2000), p. 567. An extension to the exponential case of the Erdos-Mordell inequality in a triangle.

Let P be a point in the interior of ABC. Let x, y, z be distances from P to the vertices A, B, C, respectively and let q, r, t  be the distances to the sides BC, CA, AB, respectively.  Prove that for all we have the inequalities:

(1)

. (2)

5. Problem 10418, 101 (1994), 1013.

Given an acute triangle, let denote, respectively, its altitudes, and let p denote its semiperimeter. Show that

.