A problem on Algebra!!

[swastik pramanik]

Let \(a\), \(b\) and \(c\) be such that \(a+b+c=0\) and

\(\begin{align}P=\dfrac{a^{2}}{2a^{2}+bc}+\dfrac{b^{2}}{2b^{2}+ca}+\dfrac{c^{2}}{2c^{2}+ab}\end{align}\)

is defined. What is the value of \(P\)?

[abhishek sinha]

Note \(2a^2+bc= a^2-a(b+c)+bc= -(a-b)(c-a)\). Similarly, \(2b^2+ca=-(a-b)(b-c)\) and \(2c^2+ab=-(b-c)(c-a)\). Hence,

\[ P= - \frac{a^2(b-c)+b^2(c-a)+c^2(a-b)}{(a-b)(b-c)(c-a)}=1.\]

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