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Algebra Difficulty 5.7 AIME, harder Prove it

2. Prove that there do not exist such values of mm and nn for which the polynomials 3m2+4mn−2n23 m^{2}+4 m n-2 n^{2} and −m2−4mn+3n2-m^{2}-4 m n+3 n^{2} simultaneously take negative values.

Solution

2. Solution. The sum of the given polynomials is 3m2+4mn−2n2−m2−4mn+3n2=2m2+n2⩾03 m^{2}+4 m n-2 n^{2}-m^{2}-4 m n+3 n^{2}=2 m^{2}+n^{2} \geqslant 0,

i.e., it cannot be a negative number. This is only possible when at least the value of one of the polynomials is non-negative for the given values of mm and nn.

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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.