BMO 1991

If $x^2+y^2-x$ is a multiple of $2xy$ where x and y are integers then prove x is a perfect square.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

So we are given that $x^2+y^2-x-2kxy=0.$ Write $x=a^2b$ with $b$ square-free. Then $a^4b^2+y^2-a^2b-2ka^2by=0,$ so $a^2\mid y^2$ and $a\mid y$ , say $y=ac$ Then $a^2b^2+c^2-b-2kabc=0,$ so $b\mid c^2$ and as $b$ is square-fee in fact $b\mid c$ , say $c=bd$ . Then $a^2b+bd^2-1-2kabd=0,$ so $b\mid 1$ . Assume $b=-1$ . Then we have $a^2+d^2=2kad-1.$ As the right hand side is odd, exactly one of $a,d$ must be odd, the other even. But then the right hand side is $\equiv -1\pmod 4$ and the left is $\equiv +1\pmod 4$ We conclude $b\ne -1$ , hence $b=+1$ and $x=a^2.$ and the proof is completed

Refaat M. Sayed - 5 years, 10 months ago

Oh that's very nice! Much simpler than the approach I would have taken.

Calvin Lin Staff - 5 years, 9 months ago

Thanks for taking your time to it, but do you mind giving some clarification.How do you know that x is not prine or a product of primes per example 5 ,6,15,13 etc and cannot be written as you stated in your solution. Forgive me if im missing some important clue. Also i forgot to state that x and y are POSITIVE INTEGERS. Disregard this reply if you were based heavily on this conditiom.

Lorenc Bushi - 5 years, 10 months ago

See Vieta Root Jumping .

Calvin Lin Staff - 5 years, 9 months ago

Thanks,sir

Lorenc Bushi - 5 years, 9 months ago

Can you please give the main idea?I have been trying it for 3 days.

Lorenc Bushi - 5 years, 9 months ago

Also , 8 is a factor of 16 but 8 is not a factor of 4 and 8 is square free.

Lorenc Bushi - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$