23rd Putnam 1962

A1.  5 points lie in a plane, no 3 collinear. Show that 4 of the points form a convex quadrilateral.
A2.  Let R be the reals. Find all f : K → R, where K is [0, ∞) or a finite interval [0, a), such that (1/k ∫0k f(x) dx )2= f(0) f(k) for all k in K.
A3.  ABC is a triangle and k > 0. Take A' on BC, B' on CA, C' on AB so that AB' = k B'C, CA' = k A'B, BC' = k C'A. Let the three points of intersection of AA', BB', CC' be P, Q, R. Show that the area PQR (k2 + k + 1) = area ABC (k - 1)2.
A4.  R is the reals. [a, b] is an interval with b ≥ a + 2. f : [a, b] → R is twice differentiable and |f(x)| ≤ 1 and |f ''(x)| ≤ 1. Show that |f '(x)| ≤ 2.
A5.  Find nC1 12 + nC2 22 + nC3 32 + ... + nCn n2 (where nCr is the binomial coefficient).
A6.  X is a subset of the rationals which is closed under addition and multiplication. 0 ∉ X. For any rational x ≠ 0, just one of x, -x ∈ X. Show that X is the set of all positive rationals.
B1.  Define x(n) = x(x - 1)(x - 2) ... (x - n + 1) and x(0) = 1. Show that (x + y)(n) = nC0 x(0)y(n) + nC1 x(1)y(n-1) + nC2 x(2)y(n-2) + ... + nCn x(n)y(0).
B2.  Let R be the reals, let N be the set of positive integers, and let P = {X : X ⊆ N}. Find f : R → P such that f(a) ⊂ f(b) (and f(a) ≠ f(b) ) if a < b.
B3.  Show that a convex open set in the plane containing the point P, but not containing any ray from P, must be bounded. Is this true for any convex set in the plane?
B4.  A finite set of circles divides the plane into regions. Show that we can color the plane with two colors so that no two adjacent regions (with a common arc of non-zero length forming part of each region's boundary) have the same color.
B5.  Show that for n > 1, (3n + 1)/(2n + 2) < ∑1n rn/nn < 2.
B6.  f : [0, 2π) → [-1, 1] satisfies f(x) = ∑0n (aj sin jx + bj cos jx) for some real constants aj, bj. Also |f(x)| = 1 for just 2n distinct values in the interval. Show that f(x) = cos(nx + k) for some k.

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