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Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 21, 2011"

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<math>x=2y</math>
 
<math>x=2y</math>
  
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Therefore:
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Then we just plug this back into the original parameter, where <math>z=1002001</math>.
  
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Sample solution: <math>x=2, y=1</math>
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<math>a(2y)+by=1002001</math>
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<math>2ay+by=1002001</math>
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<math>y(2a+b)=1002001</math>
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<math>y=\frac{1002001}{2a+b}</math>
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<math>=\frac{143}{53}</math>
  
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General Solution: <math>x=2n, y=n</math>
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Then this also means that <math>x=\frac{286}{106}</math>

Revision as of 16:02, 21 July 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 $z=1002001$.

$a(2y)+by=1002001$ $2ay+by=1002001$ $y(2a+b)=1002001$ $y=\frac{1002001}{2a+b}$ $=\frac{143}{53}$

Then this also means that $x=\frac{286}{106}$

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