1994 USAMO Problems
Problem 1
Let 
be positive integers, no two consecutive, and let 
for 
. Prove that, for each positive integer 
the interval 
contains at least one perfect square.
Problem 2
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, red, yellow, blue?
Problem 3
A convex hexagon 
is inscribed in a circle such that 
and diagonals 
, 
, and 
are concurrent. Let 
be the intersection of and 
. Prove that 
.
Problem 4
Let 
be a sequence of positive real numbers satisfying 
for all 
. Prove that, for all 
![\[\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right).\]](http://latex.artofproblemsolving.com/8/2/9/8293056937b2c93fb3ccfde0a9b743d24fd9660d.png)
Problem 5
Let 
and 
denote the number of elements, the sum, and the product, respectively, of a finite set 
of positive integers. (If is the empty set, 
.) Let 
be a finite set of positive integers. As usual, let 
denote 
. Prove that
![\[\sum_{U \subseteq S} ( - 1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S)\]](http://latex.artofproblemsolving.com/1/c/e/1ce32fad43e7d84e34f61e05e255cfd303dc280b.png)
for all integers 
.
Resources
| 1994 USAMO (Problems • Resources) | ||
| Preceded by 1993 USAMO |
Followed by 1995 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
