7th Putnam 1947

A1.  The sequence an of real numbers satisfies an+1 = 1/(2 - an). Show that limn→∞an = 1.
A2.  R is the reals. f : R → R is continuous and satisfies f(r) = f(x) f(y) for all x, y, where r = √(x2 + y2). Show that f(x) = f(1) to the power of x2.
A3.  ABC is a triangle and P an interior point. Show that we cannot find a piecewise linear path K = K1K2 ... Kn (where each KiKi+1 is a straight line segment) such that: (1) none of the Ki do not lie on any of the lines AB, BC, CA, AP, BP, CP; (2) none of the points A, B, C, P lie on K; (3) K crosses each of AB, BC, CA, AP, BP, CP just once; (4) K does not cross itself.
A4.  Take the x-axis as horizontal and the y-axis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity v. Assuming the only force on the pellet after firing is gravity (acceleration g), which points in the first quadrant can the gun hit?
A5.  The sequences an, bn, cn of positive reals satisfy: (1) a1 + b1 + c1 = 1; (2) an+1 = an2 + 2bncn, bn+1 = bn2 + 2cnan, cn+1 = cn2 + 2anbn. Show that each of the sequences converges and find their limits.
A6.  A is the matrix
a  b  c

d  e  f

g  h  i

det A = 0 and the cofactor of each element is its square (for example the cofactor of b is fg - di = b2). Show that all elements of A are zero.
B1.  Let R be the reals. f : [1, ∞) → R is differentiable and satisfies f '(x) = 1/(x2 + f(x)2) and f(1) = 1. Show that as x → ∞, f(x) tends to a limit which is less than 1 + π/4.
B2.  R is the reals. f :(0, 1) → R is differentiable and has a bounded derivative: |f '(x)| <= k. Prove that : |∫01 f(x) dx - ∑1n f(i/n) /n| ≤ k/n.
B3.  Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2?
B4.  p(z) = z2 + az + b has complex coefficients. |p(z)| = 1 on the unit circle |z| = 1. Show that a = b = 0.
B5.  Let p(x) be the polynomial (x - a)(x - b)(x - c)(x - d). Assume p(x) = 0 has four distinct integral roots and that p(x) = 4 has an integral root k. Show that k is the mean of a, b, c, d.
B6.  P is a variable point in space. Q is a fixed point on the z-axis. The plane normal to PQ through P cuts the x-axis at R and the y-axis at S. Find the locus of P such that PR and PS are at right angles.

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
5 Mar 2002