Maths Olympiad Prep

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Algebra Difficulty 6.1 National olympiad Prove it

104. Let a,b,c,da, b, c, d be real numbers. Prove the inequality: (a+b+c+d)2⩽3(a2+b2+c2+(a+b+c+d)^{2} \leqslant 3\left(a^{2}+b^{2}+c^{2}+\right. d2)+6ab.(−1998\left.d^{2}\right)+6 a b .(-1998 Polish Mathematical Olympiad problem)

Solution

104. By the Cauchy-Schwarz inequality, we have [(a+b)+c+d)]2⩽3[(a+b)2+c2+d2][(a+b)+c+d)]^{2} \leqslant 3\left[(a+b)^{2}+c^{2}+d^{2}\right], rearranging yields. □\square

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