1993 USAMO Problems

Problem 1

For each integer $n\ge2$ , determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying

$a^n = a + 1, \quad b^{2n} = b + 3a$

is larger.

Problem 2

Let $ABCD$ be a convex quadrilateral such that diagonals $AC$ and $BD$ intersect at right angles, and let $E$ be their intersection. Prove that the reflections of $E$ across $AB$ , $BC$ , $CD$ , $DA$ are concyclic.

Solution

Problem 3

Consider functions $f : [0, 1] \rightarrow \Re$ which satisfy

	(i)	$f(x)\ge0$ for all $x$ in $[0, 1]$ ,
	(ii)	$f(1) = 1$ ,
	(iii)	$f(x) + f(y) \le f(x + y)$ whenever $x$ , $y$ , and $x + y$ are all in $[0, 1]$ .

Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$

for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$ .

Solution

Problem 4

Let $a$ , $b$ be odd positive integers. Define the sequence $(f_n)$ by putting $f_1 = a$ , $f_2 = b$ , and by letting fn for $n\ge3$ be the greatest odd divisor of $f_{n-1} + f_{n-2}$ . Show that $f_n$ is constant for $n$ sufficiently large and determine the eventual value as a function of $a$ and $b$ .

Solution

Problem 5

Let $a_0, a_1, a_2,\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\le a^2_i$ for $i = 1, 2, 3,\cdots$ . (Such a sequence is said to be log concave.) Show that for each $n > 1$ ,