24th Putnam 1963

A1.  Dissect a regular 12-gon into a regular hexagon, 6 squares and 6 equilateral triangles. Let the regular 12-gon have vertices P1, P2, ... , P12 (in that order). Show that the diagonals P1P9, P12P4 and P2P11 are concurrent.
A2.  The sequence a1, a2, a3, ... of positive integers is strictly monotonic increasing, a2 = 2 and amn = aman for m, n relatively prime. Show that an = n.
A3.  Let D be the differential operator d/dx and E the differential operator xD(xD - 1)(xD - 2) ... (xD - n). Find an expression of the form y = ∫1x g(t) dt for the solution of the differential equation Ey = f(x), with initial conditions y(1) = y'(1) = ... = y(n)(1) = 0, where f(x) is a continuous real-valued function on the reals.
A4.  Show that for any sequence of positive reals, an, we have lim supn→∞ n( (an+1 + 1)/an - 1) ≥ 1. Show that we can find a sequence where equality holds.
A5.  R is the reals. f : [0, π] → R is continuous and ∫0π f(x) sin x dx = ∫0π f(x) cos x dx = 0. Show that f is zero for at least two points in (0, π). Hence or otherwise, show that the centroid of any bounded convex open region of the plane is the midpoint of at least three distinct chords of its boundary.
A6.  M is the midpoint of a chord PQ of an ellipse. A, B, C, D are four points on the ellipse such that AC and BD intersect at M. The lines AB and PQ meet at R, and the lines CD and PQ meet at S. Show that M is also the midpoint of RS.
B1.  Find all integers n for which x2 - x + n divides x13 + x + 90.
B2.  Is the set { 2m3n: m, n are integers } dense in the positive reals?
B3.  R is the reals. Find all f : R → R which are twice differentiable and satisfy: f(x)2 - f(y)2 = f(x + y) f(x - y).
B4.  Γ is a closed plane curve enclosing a convex region and having a continuously turning tangent. A, B, C are points of Γ such that ABC has the maximum possible perimeter p. Show that the normals to Γ at A, B, C are the angle bisectors of ABC. If A, B, C have this property, does ABC necessarily have perimeter p? What happens if Γ is a circle?
B5.  The series ∑ an of non-negative terms converges and ai ≤ 100an for i = n, n + 1, n + 2, ... , 2n. Show that limn→∞ nan = 0.
B6.  Let S = S0 be a set of points in space. Let Sn = { P : P belongs to the closed segment AB, for some A, B ∈ Sn-1}. Show that S2 = S3.

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
6 Feb 2002