9th Putnam 1949

A1.  Do either (1) or (2)
(1)   Let L be the line through (0, -a, a) parallel to the x-axis, M the line through (a, 0, -a) parallel to the y-axis, and N the line through (-a, a, 0) parallel to the z-axis. Find the equation of S, the surface formed from the union of all lines K which intersect each of L, M and N.
(2)   Let S be the surface xy + yz + zx = 0. Which planes cut S in circles? In parabolas?
A2.  Take points O, P, Q, R in space. Let the volume of the parallelepiped with edges OP, OQ, OR be V. Let V' be the volume of the parallelepiped which has O as one vertex and which has OP, OQ, OR as altitudes to three faces. Show that V V' = OP2OQ2OR2. Generalize to n dimensions.
A3.  All the complex numbers zn are non-zero and |zm - zn| > 1 (for any m ≠ n). Show that ∑ 1/zn3 converges.
A4.  Take P inside the tetrahedron ABCD to minimize PA + PB + PC + PD. Show that ∠APB = ∠CPD and that the bisector of APB also bisects CPD.
A5.  Let p(z) = z6 + 6z + 10. How many roots lie in each quadrant of the complex plane?
A6.  Show that ∏1 (1 + 2 cos(2z/3n)/3 = (sin z)/z for all complex z.
B1.  Show that for any rational a/b ∈ (0, 1), we have |a/b - 1/√2| > 1/(4b2).
B2.  Do either (1) or (2)
(1)   Prove that ∑2 cos(ln ln n) / ln n diverges.
(2)   Let k, a, b, c be real numbers such that a, k > 0 and b2 < ac. Show that ∫U (k + ax2 + 2bxy + cy2)-2 dx dy = π/( k √(ac - b2) ), where U is the entire plane.
B3.  C is a closed plane curve. If P, Q ∈ C, then |PQ| < 1. Show that we can find a disk radius 1/√3 which contains C.
B4.  Let (1 + x - √(x2 - 6x + 1) )/4 = ∑1 anxn. Show that all an are positive integers.
B5.  an is a sequence of positive reals. Show that lim supn→∞( (a1 + an+1)/an)n ≥ e.
B6.  C is a closed convex curve. If P lies on C and TP is the tangent at P, then TP varies continuously with P. Let O be a point inside C. Given a point P on C, define f(P) to be the point where the perpendicular from O to TP intersects C. Given P1, define the sequence Pn by Pn+1 = f(Pn). Assume that f is continuous and that, for each P, C lies entirely on one side of TP. Show that Pn converges. Find S = { P : P = limn→∞Pn for some P1}.

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