Go Back > Fun Stuff > Puzzles

Thread Tools
Old 2007-03-24, 00:55   #1
fetofs's Avatar
Aug 2005

36210 Posts
Default 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.
fetofs is offline   Reply With Quote
Old 2007-03-24, 15:46   #2
R. Gerbicz
R. Gerbicz's Avatar
"Robert Gerbicz"
Oct 2005

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<n but this is a contradiction.
R. Gerbicz is offline   Reply With Quote

Thread Tools

Similar Threads
Thread Thread Starter Forum Replies Last Post
What are the odds? petrw1 PrimeNet 0 2016-10-06 22:40
Odds Fred Software 4 2016-03-08 03:05
ECM odds westicles Miscellaneous Math 4 2015-05-25 22:04
Seems to defy the odds.... petrw1 Factoring 6 2013-03-19 00:21
odds in genetics. science_man_88 Science & Technology 10 2010-11-09 22:01

All times are UTC. The time now is 23:23.

Wed Nov 25 23:23:43 UTC 2020 up 76 days, 20:34, 3 users, load averages: 1.29, 1.40, 1.35

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.