1993 USAMO Problems
Problem 1
For each integer , determine, with proof, which of the two positive real numbers
and
satisfying
![$a^n = a + 1, \quad b^{2n} = b + 3a$](http://latex.artofproblemsolving.com/5/c/8/5c8cac81e3d9c2751d3306d933e8d4212fa64c08.png)
is larger.
Problem 2
Let be a convex quadrilateral such that diagonals
and
intersect at right angles, and let
be their intersection. Prove that the reflections of
across
,
,
,
are concyclic.
Problem 3
Consider functions which satisfy
(i) | ![]() ![]() ![]() | |
(ii) | ![]() | |
(iii) | ![]() ![]() ![]() ![]() ![]() |
Find, with proof, the smallest constant such that
for every function satisfying (i)-(iii) and every
in
.
Problem 4
Let ,
be odd positive integers. Define the sequence
by putting
,
, and by letting fn for
be the greatest odd divisor of
.
Show that
is constant for
sufficiently large and determine the eventual
value as a function of
and
.
Problem 5
Let be a sequence of positive real numbers satisfying
for
. (Such a sequence is said to be log concave.) Show that for
each
,
![$\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}$](http://latex.artofproblemsolving.com/d/a/2/da21136d8f04ee35b3510f5fb8653b483dc8131b.png)