1979 USAMO Problems/Problem 3: Difference between revisions
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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>. | ||
==Solution== | |||
{{solution}} | |||
==See also== | |||
{{USAMO box|year=1979|num-b=2|num-a=4}} | {{USAMO box|year=1979|num-b=2|num-a=4}} | ||
Revision as of 22:45, 11 April 2012
Problem
is an arbitrary sequence of positive integers. A member of the sequence is picked at
random. Its value is 
. Another member is picked at random, independently of the first. Its value is 
. Then a third value, 
. Show that the probability that $a \plus{ } b \plus{ } c$ (Error compiling LaTeX. Unknown error_msg) is divisible by 
is at least 
.
Solution
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See also
| 1979 USAMO (Problems • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
