The Sixty-Second William Lowell Putnam Mathematical Competition

The 69 th William Lowell Putnam Mathematical Competition

A3 Start with a finite sequence a1, a2, . . . , an of positive integers. If possible, choose two indices j < k such that aj does not divide ak, and replace aj and ak by gcd(aj , ak) and lcm(aj , ak), respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: gcd means greatest common divisor and lcm means leas...

Solutions to the 64 th William Lowell Putnam Mathematical Competition Saturday , December 6 , 2003

The William Lowell Putnam Mathematical Competition.

Solutions to the 69 th William Lowell Putnam Mathematical Competition

A–1 The function g(x) = f (x,0) works. Substituting (x,y,z) = (0,0,0) into the given functional equation yields f (0,0) = 0, whence substituting (x,y,z) = (x,0,0) yields f (x,0) + f (0,x) = 0. Finally, substituting (x,y,z) = (x,y,0) yields f (x,y) = − f (y,0) − f (0,x) = g(x)−g(y). Remark: A similar argument shows that the possible functions g are precisely those of the form f (x,0) + c for som...

The 63 rd William Lowell Putnam Mathematical Competition Saturday

B–4 An integer n, unknown to you, has been randomly chosen in the interval [1, 2002] with uniform probability. Your objective is to select n in an odd number of guesses. After each incorrect guess, you are informed whether n is higher or lower, and you must guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of w...