1975 USAMO Problems/Problem 1

Problem

(a) Prove that

$[5x]+[5y]\ge [3x+y]+[3y+x]$,

where $x,y\ge 0$ and $[u]$ denotes the greatest integer $\le u$ (e.g., $[\sqrt{2}]=1$).

(b) Using (a) or otherwise, prove that

$\frac{(5m)!(5n)!}{m!n!(3m+n)!(3n+m)!}$

is integral for all positive integral $m$ and $n$.

Solution

 1975 USAMO (Problems • Resources) Preceded byFirst Question Followed byProblem 2 1 • 2 • 3 • 4 • 5 All USAMO Problems and Solutions