mersenneforum.org Odds and evens
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 2007-03-24, 00:55 #1 fetofs     Aug 2005 Brazil 36210 Posts Odds and evens You have three sequences A = a1, a2,...,an, B = b1, b2,...,bn and C = c1, c2,...,cn. For each 1 <= i <= n it is known that at least one of ai, bi and ci is odd. Prove that there are integers r, s and t such that rai + sbi + tci is odd for at least $\frac{4n}7$ values of i.
 2007-03-24, 15:46 #2 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 23·3·59 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

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