第 11 題
若\(x_1,x_2,x_3,x_4,x_5\)皆為實數，且滿足\(\cases{x_1x_2+x_1x_3+x_1x_4+x_1x_5=-2 \cr
x_2x_1+x_2x_3+x_2x_4+x_2x_5=-2 \cr
x_3x_1+x_3x_2+x_3x_4+x_3x_5=-2 \cr
x_4x_1+x_4x_2+x_4x_3+x_4x_5=-2 \cr
x_5x_1+x_5x_2+x_5x_3+x_5x_4=-2}\)，則\(x_1^3+x_2^3+x_3^3+x_4^3+x_5^3\)之最大值為。
[解答]
變數改成 a、b、c、d、e
ab + ac + ad + ae = -2
ba + bc + bd + be = -2
ca + cb + cd + ce = -2
da + db + dc + de = -2
ea + eb + ec + ed = -2
設 m = a + b + c + d + e
a(m - a) = -2
b(m - b) = -2
c(m - c) = -2
d(m - d) = -2
e(m - e) = -2
a(m - a) = b(m - b)
a = b 或 m = a + b，即 ab = -2
(1) 五數均不相等
則 ab = -2，bc = -2，a = c (不合)
(2) 五數均相等
則 4a^2 = -2 (不合)
(3) 恰有四數相等，令 b = c = d = e
則 4ab = -2，ab = -1/2 (不合)
(4) 有三數相等，令 c = d = e
則 ab + 3ac = -2，ab + 3bc = -2，a = b
a^2 + 3ac = -2，2ac + c^2 = -2
a^2 + 3ac = 2ac + 2c^2
a = -2c 或 a = c (不合)
4c^2 - 6c^2 = -2
c = 1 或 -1
a = -2 或 2
(a，b，c，d，e) = (2，2，-1，-1，-1) 或 (-2，-2，1，1，1)
a^3 + b^3 + c^3 + d^3 + e^3 = 13 或 -13
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