Question

设x1~x7是自然数，且x1＜x2＜...＜x7，x1+x2=x3，x2+x3=x4，x3+x4=x5，x4+x5=x6，x5+x6=x7又x1+...x7=2010

求x1+x2+x3的最大值

Answer 1

x1 + x2 + x3 = 2(x1+x2)，即求x1+x2的最大值
x3 = x1 + x2
x4 = x2 + x3 = x1 + 2x2
x5 = x3 + x4 = 2x1 + 3x2
x6 = x4 + x5 = 3x1 + 5x2
x7 = x5 + x6 = 5x1 + 8x2
x1 + ... + x7 = 13x1 + 20x2 = 2010
x2 > x1 即 x2 ≥ x1 + 1 代入得
33x2 - 13 ≥ 2010
x2 ≥ 62
x1 + x2 = x2 + (2010-20x2)/13 = (2010 - 7x2)/13
取 x2 = 62 , x1 = 770/13 不为整数
x2 = 63 , x1 = 750/13 不为整数
代入得 x2 = 68 时，x1 = 50 为整数
∴x1 + x2 + x3的最大值为 2(x1+x2) = 236

Answer 2

x1+...x7=2010
13x1+20x2=2010
求x1+x2+x3=2(x1+x2)的最大值
13x1+20x2=2010写成16.5(x1+x2)+3.5(x2-x1)=2010........(I)
x2-x1≥1
16.5(x1+x2)≤2010-3.5=2006.5
x1+x2≤121
注意x1，x2都是自然数，所以x1+x2=121不一定能满足条件
①x1+x2=121，代入(I)发现x2-x1不是整数
②x1+x2=120，代入(I)...
③x1+x2=119，....
④x1+x2=118，发现x2-x1=18，那么x1=50，x2=68是满足条件，并且使x1+x2最大