※ 引述《lovewin (我要當102年榜首)》之銘言：
: D 8. 將14 分成若干個正整數的和，並使這些數的乘積值最大，則最大的積為何？
: (A) 49 (B) 96 (C) 128(D) 162 101實中
14 is expressed as the sum of n positive real numbers
a_1 + a_2 + ... + a_n =14
then
a_1 * a_2 * .... * a_n <= (14/n)^n
(14/1)^1=14, (14/2)^2=49, (14/3)^3<102 (14/4)^4<151
(14/5)^5<173, (14/6)^6<162, (14/7)^7=128,...
We guess n=5,
a<=b<=c<=d<=e, a+b+c+d+e=14
5a<=14, a<=2.8 and a is a positive integer.
(i) a=1, b+c+d+e=13, abcde=bcde<=(13/4)^4<112
(ii) a=2, b+c+d+e=12, abcde=2bcde<=2(12/4)^4=162
and max occurs at a=2,b=c=d=e=3
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Furthermore, 162 is still max for all n.
We do not need to check other n.
Done.
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