AoPS Wiki talk:Problem of the Day/July 21, 2011

Problem

AoPSWiki:Problem of the Day/July 21, 2011

Solution

Note: someone check the arithmetic please

For purposes of generalization, let the equations be $ax+by=1002001$ and $cx+dy=2004002$. Notice that $2*1002001 = 2004002$. Swap this out for a new variable $z$. This gives $ax+by=z, cx+dy=2z$.

Multiply the left equation by two and substitute it into the other equation.

$2ax+2by=cx+dy$, which implies that$(c-2a)x=(2b-d)y$. Substituting the actual numbers back in gives: $(997997-686686)x=(630630-8008)y$ Simplifying: $311311x=622722y$ Which further simplifies to: $x=2y$

Then we just plug this back into the original parameter, where $z=1002001$.

And we find that the only solution is $x=2, y=1$

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