54th Putnam 1993

A1.  Let O be the origin. y = c intersects the curve y = 2x - 3x3 at P and Q in the first quadrant and cuts the y-axis at R. Find c so that the region OPR bounded by the y-axis, the line y = c and the curve has the same area as the region between P and Q under the curve and above the line y = c.
A2.  The sequence an of non-zero reals satisfies an2 - an-1an+1 = 1 for n ≥ 1. Prove that there exists a real number a such that an+1 = a an - an-1 for n ≥ 1.
A3.  Let P be the set of all subsets of {1, 2, ... , n}. Show that there are 1n + 2n + ... + mn functions f : P → {1, 2, ... , m} such that f(A ∩ B) = min( f(A), f(B) ) for all A, B.
A4.  Given a sequence of 19 positive (not necessarily distinct) integers not greater than 93, and a set of 93 positive (not necessarily distinct) integers not greater than 19. Show that we can find non-empty subsequences of the two sequences with equal sum.
A5.  Let U be the set formed as the union of three open intervals, U = (-100, -10) ∪ (1/101, 1/11) ∪ (101/100, 11/10). Show that ∫U (x2 - x)2/(x3 - 3x + 1)2 dx is rational.
A6.  Let a0, a1, a2, ... be a sequence such that: a0 = 2; each an = 2 or 3; an = the number of 3s between the nth and n+1th 2 in the sequence. So the sequence starts: 233233323332332 ... . Show that we can find α such that an = 2 iff n = [αm] for some integer m ≥ 0.
B1.  What is the smallest integer n > 0 such that for any integer m in the range 1, 2, 3, ... , 1992 we can always find an integral multiple of 1/n in the open interval (m/1993, (m + 1)/1994)?
B2.  A deck of 2n cards numbered from 1 to 2n is shuffled and n cards are dealt to A and B. A and B alternately discard a card face up, starting with A. The game when the sum of the discards is first divisible by 2n + 1, and the last person to discard wins. What is the probability that A wins if neither player makes a mistake?
B3.  x and y are chosen at random (with uniform density) from the interval (0, 1). What is the probability that the closest integer to x/y is even?
B4.  K(x, y), f(x) and g(x) are positive and continuous for x, y ∈ [0, 1]. ∫01 f(y) K(x, y) dy = g(x) and ∫01 g(y) K(x, y) dy = f(x) for all x ∈ [0, 1]. Show that f = g on [0, 1].
B5.  Show that given any 4 points in the plane we can find two whose distance apart is not an odd integer.
B6.  Given a triple of positive integers x ≤ y ≤ z, derive a new triple as follows. Replace x and y by 2x and y - x (and reorder), or replace x and z by 2x and z - x (and reorder), or replace y and z by 2y and z - y (and reorder). Show that by finitely many tranformations of this type we can always derive a triple with smallest element zero.

To avoid possible copyright problems, I have changed the wording, but not the substance of all the problems. The original text of the problems and the official solutions are in American Mathematical Monthly 101 (1994) 727.  
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© John Scholes
6 Jan 2001