1992 Canada National Olympiad
1
Prove that the product of the first 
natural numbers is divisible by the sum of the first natural numbers if and only if 
is not an odd prime.
natural numbers is divisible by the sum of the first natural numbers if and only if 
is not an odd prime.
establish the inequality![\[ x(x-z)^2 + y(y-z)^2 \geq (x-z)(y-z)(x+y-z)\]](http://latex.artofproblemsolving.com/3/4/3/343ac816107220101cd3601e5986914f4771dfb9.png)
3
In the diagram, 
is a square, with 
and 
interior points of the sides 
and 
respectively. Determine all the possible ways of selecting and so as to maximize the area of the quadrilateral 
.
is a square, with 
and 
interior points of the sides 
and 
respectively. Determine all the possible ways of selecting and so as to maximize the area of the quadrilateral 
.
![\[ x^2 + \frac{x^2}{(x+1)^2} = 3\]](http://latex.artofproblemsolving.com/b/2/e/b2ea8d02ef28232f648bb60d1d8df6b6e4711d84.png)
