Problem of the Week

October 18

Find all prime numbers p such that 13p + 1 is a perfect square.

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, October 22.

October 4

Consider the following infinite array of positive integers.

4 7 10 13 16 19 ...
7 12 17 22 27 32 ...
10 17 24 31 38 45 ...
13 22 31 40 49 58 ...
:

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.

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, October 8.
(Problem #4 will appear on the Friday after midterm break.)

Congratulations to solver: Rosica Dineva (06).


September 27

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?

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, October 1.

Congratulations to solvers: Rosica Dineva (06), Alina Florescu (06), & Ha Le (05).


September 20

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.

Congratulations to solvers: Ha Le (05), Jen Leahy (05), Patricia Thomas (06), & Rosy Dineva (06).