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Algebra Difficulty 5.0 AIME Find the answer

3. If the real numbers a,b,c,d,ea, b, c, d, e satisfy the conditions
a+b+c+d+e=8,a2+b2+c2+d2+e2=16. \begin{array}{l} a+b+c+d+e=8, \\ a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16 . \end{array}

Determine the maximum value of ee.

A number or a short expression. Spacing and $ signs are ignored.

Solution

f(x)=4x2+2(a+b+c+d)x+(a2+b2+c2+d2)=(x+a)2+(x+b)2+(x+c)2+(x+d)2⩾0. \begin{array}{l} f(x)=4 x^{2}+2(a+b+c+d) x+\left(a^{2}+b^{2}+c^{2}+d^{2}\right) \\ =(x+a)^{2}+(x+b)^{2}+(x+c)^{2}+(x+d)^{2} \\ \geqslant 0 . \end{array}

Therefore, Δ⩽0\Delta \leqslant 0. Solving this, we get 0⩽e⩽1650 \leqslant e \leqslant \frac{16}{5}.

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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.