Problems

4584S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Joe Dan Austin, Houston, TX.
The usual formula for the area of a triangle, A = (1/2)bh, where the base b is a side of the triangle and h is the corresponding altitude, presupposes that the formula yields the same value for A, regardless of which side is taken as the base b. Prove that this is in fact the case.

4585: Proposed by Richard L. Francis, Cape Girardeau, MO.
A pair of integers {x, x + 2} in which one element is prime and the other composite is called discordant. Show that there are infinitely many discordant pairs with the prime as the greater element (prime dominant) and infinitely many in which the composite is the greater (composite dominant).

4586: Proposed by John P. Hoyt, Lancaster, PA.
A quadrilateral has consecutive sides of constant lengths a, b, c and d. The longest side has length a and a + c = b + d. The endpoints of the longest side are fixed while the other two vertices are free to move about. Determine the locus of the center of the inscribed circle.

4587: Proposed by Murray S. Klamkin, Edmonton, AB, Canada.
Let ABC be an acute-angled, non-equilateral triangle. Prove that there is an interior point P other than the circumcenter such that

4588: Proposed by Marty Sambe-rg and Andrew Cusumano, Gmat Neck, NY.

4589: Proposed by Monte J. Zerger, Alamosa, CO.
(a) Find infinitely many pairs of triangular numbers such that the sum of each pair, as well as the sum of the squares of each pair is (al) a triangular number (a2) a square number.
(b) Consider the set S of natural numbers that can be written as both the sum of two triangular numbers and the sum of the squares of two triangular numbers. Show that infinitely many elements of S are (bl) triangular numbers (b2) square numbers.