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 P_a be the parabola y = a³x²/3 + a²x/2 - 2a. Find the locus of the vertices of P_a, and the envelope of P_a. Sketch the envelope and two P_a.
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/r^k 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.