20070324, 00:55  #1 
Aug 2005
Brazil
2×181 Posts 
Odds and evens
You have three sequences A = a_{1}, a_{2},...,a_{n}, B = b_{1}, b_{2},...,b_{n} and C = c_{1}, c_{2},...,c_{n}. For each 1 <= i <= n it is known that at least one of a_{i}, b_{i} and c_{i} is odd. Prove that there are integers r, s and t such that ra_{i} + sb_{i} + tc_{i} is odd for at least values of i.

20070324, 15:46  #2 
"Robert Gerbicz"
Oct 2005
Hungary
17·83 Posts 
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. 
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