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2006 Cyprus Seniors Provincial/2nd grade/Problem 4

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Problem

Find all integers pairs (x,y) that verify at the same time the inequalities x^2\leq\frac{y^2+2x-1}{2} and y^2\leq\frac{x^2-2y-1}{2}.


Solution

The inequality x^2\leq\frac{y^2+2x-1}{2} becomes 2x^2\leq y^2+2x-1 and the inequality y^2\leq\frac{x^2-2y-1}{2} becomes 2y^2\leq x^2-2y-1

By adding the inequalities we have 2x^2+2y^2 \leq y^2+2x-1+x^2-2y-1

x^2-2x+1+2y^2+2y+1 \leq 0

(x-1)^2+(y+1)^2 \leq 0

But always (x-1)^2+(y+1)^2 \geq 0

So (x-1)^2+(y+1)^2 = 0

And the only integer pair is (1,-1)


See also

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