变量为整数的不等式

jhsinutopia

$\{a,b,c,d,e\}=\{1,2,3,4,5\}$
$max \left(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{e}+\dfrac{e}{a}\right)=?$

ZCos666

n	排列	最大值
1	1	$1$
2	1, 2	$\dfrac{5}{2}$
3	1, 2, 3	$\dfrac{25}{6}$
4	1, 3, 2, 4	$\dfrac{19}{3}$
5	1, 4, 2, 3, 5	$\dfrac{511}{60}$
6	1, 5, 2, 4, 3, 6	$\dfrac{331}{30}$
7	1, 6, 2, 5, 4, 3, 7	$\dfrac{1901}{140}$
8	1, 6, 3, 5, 4, 7, 2, 8	$\dfrac{3431}{210}$
9	1, 7, 3, 6, 5, 4, 8, 2, 9	$\dfrac{24127}{1260}$
10	1, 9, 2, 7, 4, 6, 5, 8, 3, 10	$\dfrac{11141}{504}$
11	1, 10, 2, 8, 4, 7, 6, 5, 9, 3, 11	$\dfrac{348113}{13860}$
12	1, 11, 2, 9, 4, 8, 6, 7, 5, 10, 3, 12	$\dfrac{97841}{3465}$
13	1, 11, 3, 9, 5, 8, 7, 6, 10, 4, 12, 2, 13	$\dfrac{1257761}{40040}$
14	1, 12, 3, 10, 5, 9, 7, 8, 6, 11, 4, 13, 2, 14	$\dfrac{12496903}{360360}$
15	1, 13, 3, 11, 5, 10, 7, 8, 9, 6, 12, 4, 14, 2, 15	$\dfrac{2738339}{72072}$
16	1, 15, 2, 13, 4, 11, 6, 10, 8, 9, 7, 12, 5, 14, 3, 16	$\dfrac{29828243}{720720}$
17	1, 16, 2, 14, 4, 12, 6, 11, 8, 9, 10, 7, 13, 5, 15, 3, 17	$\dfrac{109841015}{2450448}$