Odds and evens
 

You have three sequences A = a[sub]1[/sub], a[sub]2[/sub],...,a[sub]n[/sub], B = b[sub]1[/sub], b[sub]2[/sub],...,b[sub]n[/sub] and C = c[sub]1[/sub], c[sub]2[/sub],...,c[sub]n[/sub]. For each 1 <= i <= n it is known that at least one of a[sub]i[/sub], b[sub]i[/sub] and c[sub]i[/sub] is odd. Prove that there are integers [I]r[/I], [I]s[/I] and [I]t[/I] such that ra[sub]i[/sub] + sb[sub]i[/sub] + tc[sub]i[/sub] is odd for at least [tex]\frac{4n}7[/tex] values of i.

[spoiler]
pattern number of combinations:
of (a%2,b%2,c%2):
1,1,1 p7
1,1,0 p6
1,0,1 p5
1,0,0 p4
0,1,1 p3
0,1,0 p2
0,0,1 p1

We know that: p1+p2+p3+p4+p5+p6+p7=n

Indirectly suppose that there is no good r,s,t integer values


Let r=0,s=0,t=1, is bad if and only if
p1+p3+p5+p7<4/7*n is true. Similarly:
Let r=0,s=1,t=0, then p2+p3+p6+p7<4/7*n
Let r=0,s=1,t=1, then p1+p2+p5+p6<4/7*m
Let r=1,s=0,t=0, then p4+p5+p6+p7<4/7*n
Let r=1,s=0,t=1, then p1+p3+p4+p6<4/7*n
Let r=1,s=1,t=0, then p2+p3+p4+p5<4/7*n
Let r=1,s=1,t=1, then p1+p2+p4+p7<4/7*n
Add these 7 inequalities:
4*(p1+p2+p3+p4+p5+p6+p7)<4*n So:

p1+p2+p3+p4+p5+p6+p7<n but this is a contradiction.
[/spoiler]