21st Putnam 1960

A1.  For n a positive integer find f(n), the number of pairs of positive integers (a, b) such that ab/(a + b) = n.
A2.  Let S be the set consisting of a square with side 1 and its interior. Show that given any three points of S, we can find two whose distance apart is at most √6 - √2.
A3.  Let a, b, g, d, e be arbitary reals. Show that (1 - α) eα + (1 - β) eα+β + (1 - γ) eα+β+γ + (1 - δ) eα+β+γ+δ + (1 - ε) eα+β+γ+δ+ε ≤ k4, where k1 = e, k2 = k1e, k3 = k2e, k4 = k3e (so k4 is 10k with k approx 1.66 million).
A4.  Given two points P, Q on the same side of a line l, find the point X which minimises the sum of the distances from X to P, Q and l.
A5.  The real polynomial p(x) is such that for any real polynomial q(x), we have p(q(x)) = q(p(x)). Find p(x).
A6.  A player throws a fair die (prob 1/6 for each of 1, 2, 3, 4, 5, 6 and each throw independent) repeatedly until his total score ≥ n. Let p(n) be the probability that his final score is n. Find lim p(n).
A7.  Let f(n) be the smallest integer such that any permutation on n elements, repeated f(n) times, gives the identity. Show that f(n) = p f(n - 1) if n is a power of p, and f(n) = f(n - 1) if n is not a prime power.
B1.  Find all pairs of unequal integers m, n such that mn = nm.
B2.  Let f(m, n) = 3m+n+(m+n)2. Find ∑00 2-f(m, n).
B3.  Fluid flowing in the plane has the velocity (y + 2x - 2x3 - 2xy2, - x) at (x, y). Sketch the flow lines near the origin. What happens to an individual particle as t → ∞ ?
B4.  Show that if an (infinite) arithmetic progression of positive integers contains an nth power, then it contains infinitely many nth powers.
B5.  Define an by a0 = 0, an+1 = 1 + sin(an - 1). Find lim (∑0n ai)/n.
B6.  Let 2f(n) be the highest power of 2 dividing n. Let g(n) = f(1) + f(2) + ... + f(n). Prove that ∑ exp( -g(n) ) converges.
B7.  Let R' be the non-negative reals. Let f, g : R' → R be continuous. Let a : R' → R be the solution of the differential equation: a' + f a = g, a(0) = c. Show that if b : R' → R satisfies b' + f b ≥ g for all x and b(0) = c, then b(x) ≥ a(x) for all x. Show that for sufficiently small x the solution of y' + f y = y2, y(0) = d, is y(x) = max ( d e-h(x) - ∫0x e-(h(x)-h(t) ) u(t)2 dt ), where the maximum is taken over all continuous u(t), and h(t) = ∫0t (f(s) - 2u(s)) ds.

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and solutions are available in: A M Gleason, R E Greenwood & L M Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA 1980. Out of print, but available in some university libraries.

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© John Scholes
20 Oct 1999