Solutions to the 70 th William Lowell Putnam Mathematical Competition Saturday , December 5 , 2009

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

Solutions to the 76 th William Lowell Putnam Mathematical Competition Saturday , December 5 , 2015

Second solution: For any λ > 0, the map (x,y) 7→ (λx,λ−1y) preserves both areas and the hyperbola xy = 1. We may thus rescale the picture so that A,B are symmetric across the line y = x, with A above the line. As P moves from A to B, the area of APB increases until P passes through the point (1,1), then decreases. Consequently, P = (1,1) achieves the maximum area, and the desired equality is ob...

Solutions to the 78th William Lowell Putnam Mathematical Competition Saturday, December 2, 2017

Remark: A similar argument shows that any secondorder linear recurrent sequence also satisfies a quadratic second-order recurrence relation. A familiar example is the identity Fn−1Fn+1 − F2 n = (−1)n for Fn the nth Fibonacci number. More examples come from various classes of orthogonal polynomials, including the Chebyshev polynomials mentioned below. Second solution. We establish directly that ...

Solutions to the 71 st William Lowell Putnam Mathematical Competition Saturday

A3 If a = b = 0, then the desired result holds trivially, so we assume that at least one of a, b is nonzero. Pick any point (a0, b0) ∈ R, and let L be the line given by the parametric equation L(t) = (a0, b0) + (a, b)t for t ∈ R. By the chain rule and the given equation, we have d dt (h◦L) = h◦L. If we write f = h◦L : R → R, then f (t) = f(t) for all t. It follows that f(t) = Ce for some consta...

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...