1975 USAMO Problems/Problem 1

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(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


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


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See Also

1975 USAMO (ProblemsResources)
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First Question
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Problem 2
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