Problem of the Week
Find all prime numbers p such that 13p + 1 is a
Consider the following infinite array of positive integers.
The array continues in such a way that each row forms an arithmetic progression, and the ith column is the same as the ith row. Show that for every positive integer n, 2n + 1 is prime if and only if n does not appear in the array.
A right circular cone has a base of radius 1 and a height of 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the length of an edge of the cube?
Draw a square with vertical and horizontal sides. Choose any point P inside the square. Connect P to each vertex of the square, dividing the square into two pairs of triangles: one pair including the two horizontal sides of the square and one pair including the two vertical sides. Show that the sum of the areas of the triangles in the first pair equals the sum of the areas of the triangles in the second pair.
Solutions should be placed in Harriet Pollatsek's mail slot in the department office or slid under her door (400 Clapp) by 5:00 p.m. Friday, September 24.