Difference between revisions of "1979 USAMO Problems/Problem 3"

(Created page with "==Problem== <math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is <math>a</math>. Another...")
 
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<math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at  
 
<math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at  
 
random. Its value is <math>a</math>. Another member is picked at random, independently of the first. Its value is <math>b</math>. Then a third value, <math>c</math>. Show that the probability that <math>a \plus{ } b \plus{ } c</math> is divisible by <math>3</math> is at least <math>\frac14</math>.
 
random. Its value is <math>a</math>. Another member is picked at random, independently of the first. Its value is <math>b</math>. Then a third value, <math>c</math>. Show that the probability that <math>a \plus{ } b \plus{ } c</math> is divisible by <math>3</math> is at least <math>\frac14</math>.
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{{USAMO box|year=1979|num-b=2|num-a=4}}

Revision as of 23:42, 11 April 2012

Problem

$a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a \plus{ } b \plus{ } c$ (Error compiling LaTeX. Unknown error_msg) is divisible by $3$ is at least $\frac14$.

1979 USAMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5
All USAMO Problems and Solutions