Algebraic Identities for RMO and IITJEE Mathematics

Below is a list of algebraic identities, which beckon a serious student of RMO or IITJEE mathematics. See if you can prove all of them. The expressions given below are symmetric and/or cyclic and/or alternating. Refer to Higher Algebra of Bernard Child if you want to refresh your theory.

$(b-c)+(c-a)+(a-b)=0$
$a(b-c)+b(c-a)+c(a-b)=0$
$a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)=-(b-c)(c-a)(a-b)$
$bc(b-c)+ca(c-a)+ab(a-b)=-(b-c)(c-a)(a-b)$
$a(b^{2}-c^{2})+b(c^{2}-a^{2})+c(a^{2}-b^{2})=(b-c)(c-a)(a-b)$
$a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)=-(b-c)(c-a)(a-b)(a+b+c)$
$(a+b+c)(bc+ca+ab)=a(b^{2}+c^{2})+b(c^{2}+a^{2})+c(a^{2}+b^{2})+3abc$
$(b+c)(c+a)(a+b)=a(b^{2}+c^{2})+b(c^{2}+a^{2})+c(a^{2}+b^{2})+2abc$
$a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-bc-ca-ab)$
$(b-c)^{2}+(c-a)^{2}+(a-b)^{2}=2(a^{2}+b^{2}+c^{2}-ab-bc-ca)$
$(a+b+c)(b+c-a)(c+a-b)(a+b-c)$ which equals

$-a^{4}-b^{4}-c^{4}+2b^{2}c^{2}+2c^{2}a^{2}+2a^{2}b^{2}$

If you have trouble proving them using the theory of symmetric/alternating expressions as elucidated in Higher Algebra of Bernard Child, please write to me and I will share my proofs with you.

A word of caution: Just because some of the expressions are simple for you to handle by brute force, please do not do so. The idea of this exercise is to underline the power of theory of symmetric expressions. I will show you more powerful examples later.

Nalin Pithwa

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