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Algebra

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17 Mar 2008 19:17:09 IST

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Quadratic

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$\text {Let} \ a,b,c,d \in \mathbb{N} \\ \\ \text {such that} \ x^2-x(a^2+b^2+c^2+d^2+1) + (ab+bc+cd+da) = 0 \ \text{has an integer root} \\ \\ \text{Prove that the other root is also an integer, and that both roots are perfect squares}$

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Hari Shankar

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18 Mar 2008 09:24:39 IST

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that's very good. nice work

sandeep ramesh

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18 Mar 2008 09:56:17 IST

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brillo pardesi

Abhijith

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18 Mar 2008 10:06:39 IST

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If the roots are m and n then

and

. It's obvious from the first equation that if m in an integer then so is n. Also since their sum and product are both positive, it follows that so are 'm' and 'n'.

Also,

from which

. But m and n are both positive, so this implies that one of them must be 1; and also that a, b, c and d are all equal.

So if we put

then

Hence, Both of them are squares.

sandeep ramesh

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18 Mar 2008 10:10:40 IST

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nice koni :)

Hari Shankar

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18 Mar 2008 11:00:25 IST

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yeah. i think you guys can stop mugging for jee and go for a holiday to singapore or mauritius

sandeep ramesh

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18 Mar 2008 11:12:46 IST

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why sirji? :P

anchit saini

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18 Mar 2008 11:21:44 IST

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because their preparation for iitjee is not good

of course it is because the prep is so good that they can enjoy now

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