43rd Putnam 1982 Problem A1 Let S be the set of points (x, y) in the plane such that |x| ≤ y ≤ |x| + 3, and y ≤ 4. Find the position of the centroid of S. Solution
: (0, 13/5). Let T be the right-angled triangle with hypoteneuse (-4, 4), (4, 4) and third vertex the origin. Let T ' be the small right-angled triangle (0, 3), (1, 4), (-1, 4). Then S is T with T ' removed. The centroid of T is (0, 8/3) and the centroid of T ' is (0, 11/3). The area of T is 16 times the area of T ', so the centroid of S is (0, y) with 15y + 11/3 = 16 8/3. Hence result. 43rd Putnam 1982 © John Scholes jscholes@kalva.demon.co.uk 3 Nov 1999