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) e^α + (1 - β) e^α+β + (1 - γ) e^α+β+γ + (1 - δ) e^α+β+γ+δ + (1 - ε) e^{α+β+γ+δ+ε} ≤ k₄, where k₁ = e, k₂ = k₁^e, k₃ = k₂^e, k₄ = k₃^e (so k₄ is 10^k 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 = n^m.
B2. Let f(m, n) = 3m+n+(m+n)². Find ∑₀^∞∑₀^∞ 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 a_n by a₀ = 0, a_n+1 = 1 + sin(a_n - 1). Find lim (∑₀ⁿ a_i)/n.
B6. Let 2^f(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) - ∫₀^x e^{-(h(x)-h(t) )} u(t)² dt ), where the maximum is taken over all continuous u(t), and h(t) = ∫₀^t (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.