Let a,b,c,d,e,f be non-negative real numbers. Suppose that a + b + c + d + e + f = 1 and
ad + be + cf >= 1/18 . Find the maximum value of ab + bc + cd + de + ef + fa.	Pick one:
A) 7/36 B) 6/36 C) 5/36 D) 4/36 E) 3/36
We partially factor   ab + bc + cd + de + ef + fa = a(b + f) + c(b + d) + e(d + f):   Notice that the terms in the parenthesis are very close to (b+d+f), except each part is missing   one term. We can add ad + cf + eb to this expression to get   a(b + f) + c(b + d)e(d + f) + ad + cf + eb   = a(b + d + f) + c(b + d + f) + e(b + d + f)   = (a + c + e)(b + d + f)   Let x = a+c+e and y = b+d+f. We have x+y = 1, so AM-GM gives us x+y   2   p   xy, hence   (a + c + e)(b + d + f) - 1   4 . Subtracting away ad + be + cf - 1   18 gives us   ab + bc + cd + de + ef + fa -   1/4   -1/18   =   7/36   where equality holds when a + c + e = b + d + f = 1   2 and ad + be + cf = 1/18 . We can verify that   equality holds when a = e = 1/9 , b = d = 1/4 ,c = 5/18, f = 0.   correct answer 7/36
<result>A</result>
Let a,b,c,d,e,f be non-negative real numbers. Suppose that a + b + c + d + e + f = 1 and
ad + be + cf >= 1/18 . Find the maximum value of ab + bc + cd + de + ef + fa.