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Algebra Difficulty 5.7 AIME, harder Prove it

12. 证明: (a,b,c)(ab,bc,ca)=(a,b)(b,c)(c,a)(a, b, c)(a b, b c, c a)=(a, b)(b, c)(c, a).

Solution

12. (a,b)(b,c)(c,a)=(a(a,b)(b,c),b(b,c)(c,a),c(c,a)(a,b))=(a,b,c)(ab,bc,ca).\begin{aligned}(a, b)(b, c)(c, a) & =(a(a, b)(b, c), b(b, c)(c, a), c(c, a)(a, b)) \\ & =(a, b, c)(a b, b c, c a) .\end{aligned}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.