[Suggested by J. Calvo] Consider a triangle ABC with sides a, b, and c, (side a is opposite the vertex A, etc., and angles alpha, beta, and gamma (alpha is the angle at A, etc.). Assume gamma is a right angle, that the length of side c=AB is given, and that the length of the bisector e of beta is given (bisector of beta is the line segment that starts at B and ends an the side b=AC, and divides the angle beta into two equal parts). Find the angle beta (in terms of lengths of c and e). For what values of c and e will there be only one solution beta? [Suggested by N. Barabanov] Consider two non-decreasing infinite sequences of positive numbers {ai} and {bi}, for i=1, 2, 3, … . Let the infinite sum of the both sequences {1/ai} and {1/bi} diverge to infinity. Does the infinite sum of the sequence {1/(ai + bi)} necessarily also diverge to infinity? (Prove or find a counter example.) [1898/2] Prove the following statement: If two triangles have a common angle, then the sum of the sines of the angles will be larger in that triangle where the difference of the remaining two angles is smaller. On the basis of this theorem, determine the shape of that triangle for which the sum of the sines of its angles is a maximum. [1894/1] Prove that the expressions 2x + 3y and 9x + 5y are divisible by 17 for the same set of integral values of x and y. Problems will be due on March 15, 2004. Check www.ndsu.edu/math/talent for solutions and more problems! Send your answers, along with this form to: Talent Search, Dept. of Mathematics, NDSU, Fargo, ND 58105. Your Name: __________________________________     Set: 1 Your Address: __________________________________ 1. _____ __________________________________ 2. _____ __________________________________ 3. _____ Your High School: __________________________________ 4. _____ Grade: _____ Total: _____