5th Putnam 1942

A1.  ABCD is a square side 2a with vertices in that order. It rotates in the first quadrant with A remaining on the positive x-axis and B on the positive y-axis. Find the locus of its center.
A2.  a and b are unequal reals. What is the remainder when the polynomial p(x) is divided (x - a)2(x - b).
A3.  Does ∑n≥0 n! kn/(n + 1)n converge or diverge for k = 19/7?
A4.  Let C be the family of conics (2y + x)2 = a(y + x). Find C', the family of conics which are orthogonal to C. At what angle do the curves of the two families meet at the origin?
A5.  C is a circle radius a whose center lies a distance b from the coplanar line L. C is rotated through π about L to form a solid whose center of gravity lies on its surface. Find b/a.
A6.  P is a plane and H is the half-space on one side of P. K is a fixed circle in P. C is a circle in P which cuts K at an angle α. Let C have center O and radius r. f(C) is the point in H on the normal to P through O and a distance r from O. Show that the locus of f(C) is a one-sheet hyperboloid and that it has two families of rulings in it.
B1.  S is a solid square side 2a. It lies in the quadrant x ≥ 0, y ≥ 0, and it is free to move around provided a vertex remains on the x-axis and an adjacent vertex on the y-axis. P is a point of S. Show that the locus of P is part of a conic. For what P does the locus degenerate?
B2.  Let Pa be the parabola y = a3x2/3 + a2x/2 - 2a. Find the locus of the vertices of Pa, and the envelope of Pa. Sketch the envelope and two Pa.
B3.  f(x, y) and g(x, y) satisfy the differential equation f1(x, y) g2(x, y) - f2(x, y) g1(x, y) = 1 (*). Taking r = f(x, y) and y as independent variables, and x = h(r, y), g(x, y) = k(r, y), show that k2(r, y) = h1(r, y). Integrate and hence obtain a solution to (*). What other solutions does (*) have?
B4.  A particle moves in a circle through the origin under the influence of a force a/rk towards the origin (where r is its distance from the origin). Find k.
B5.  Let f(x) = x/(1 + x6sin2x). Sketch the curve y = f(x) and show that ∫0 f(x) dx exists.

The Putnam fellow was Harvey Cohn. 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
4 Sep 1999
Last corrected/updated 20 January 2004