

A1. Let O be the origin. y = c intersects the curve y = 2x  3x^{3} at P and Q in the first quadrant and cuts the yaxis at R. Find c so that the region OPR bounded by the yaxis, 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 a_{n} of nonzero reals satisfies a_{n}^{2}  a_{n1}a_{n+1} = 1 for n ≥ 1. Prove that there exists a real number a such that a_{n+1} = a a_{n}  a_{n1} for n ≥ 1.


A3. Let P be the set of all subsets of {1, 2, ... , n}. Show that there are 1^{n} + 2^{n} + ... + m^{n} 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 nonempty 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} (x^{2}  x)^{2}/(x^{3}  3x + 1)^{2} dx is rational.


A6. Let a_{0}, a_{1}, a_{2}, ... be a sequence such that: a_{0} = 2; each a_{n} = 2 or 3; a_{n} = 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 a_{n} = 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]. ∫_{0}^{1} f(y) K(x, y) dy = g(x) and ∫_{0}^{1} 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.

