Difference between revisions of "1979 USAMO Problems/Problem 3"
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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 | + | 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 + b +c</math> is divisible by <math>3</math> is at least <math>\frac14</math>. |
==Solution== | ==Solution== | ||
Revision as of 18:09, 15 September 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 
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 | ||
