12th Putnam 1952

A1.  p(x) is a polynomial with integral coefficients. The leading coefficient, the constant term, and p(1) are all odd. Show that p(x) has no rational roots.
A2.  Show that the solutions of the differential equation (9 - x2) (y')2 = 9 - y2 are conics touching the sides of a square.
A3.  Let the roots of the cubic p(x) = x3 + ax2 + bx + c be α, β, γ. Find all (a, b, c) so that p(α2) = p(β2) = p(γ2) = 0.
A4.  A map represents the polar cap from latitudes -45o to 90o. The pole (latitude 90o) is at the center of the map and lines of latitude on the globe are represented as concentric circles with radii proportional to (90o - latitude). How are east-west distances exaggerated compared to north-south distances on the map at a latitude of -30o?
A5.  ai are reals ≠ 1. Let bn = 1 - an. Show that a1 + a2b1 + a3b1b2 + a4b1b2b3 + ... + anb1b2 ... bn-1 = 1 - b1b2 ... bn.
A6.  Prove that there are only finitely many cuboidal blocks with integer sides a x b x c, such that if the block is painted on the outside and then cut into unit cubes, exactly half the cubes have no face painted.
A7.  Let O be the center of a circle C and P0 a point on the circle. Take points Pn on the circle such that angle PnOPn-1 = +1 for all integers n. Given that π is irrational, show that given any two distinct points P, Q on C, the (shorter) arc PQ contains a point Pn.
B1.  ABC is a triangle with, as usual, AB = c, CA = b. Find necessary and sufficient conditions for b2c2/(2bc cos A) = b2 + c2 - 2bc cos A.
B2.  Find the surface comprising the curves which satisfy dx/(yz) = dy/(zx) = dz/(xy) and which meet the circle x = 0, y2 + z2 = 1.
B3.  Let A(x) = be the matrix
 0    a-x   b-x

-a-x   0    c-x

-b-x  -c-x   0

For which (a, b, c) does det A(x) = 0 have a repeated root in x?
B4.  The solid S consists of a circular cylinder radius r, height h, with a hemispherical cap at one end. S is placed with the center of the cap on the table and the axis of the cylinder vertical. For some k, equilibrium is stable if r/h > k, unstable if r/h < k and neutral if r/h = k. Find k and show that if r/h = k, then the body is in equilibrium if any point of the cap is in contact with the table.
B5.  The sequence an is monotonic and ∑ an converges. Show that ∑ n(an - an+1) converges.
B6.  A, B, C are points of a fixed ellipse E. Show that the area of ABC is a maximum iff the centroid of ABC is at the center of E.
B7.  Let R be the reals. Define an by a1 = α ∈ R, an+1 = cos an. Show that an converges to a limit independent of α.

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