Difference between revisions of "2023 SSMO Speed Round Problems/Problem 2"
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Thus, the answer is thus <math>\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}</math> | Thus, the answer is thus <math>\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}</math> | ||
+ | <math>1+4=\boxed{5}</math> |
Revision as of 13:24, 5 July 2023
Problem
Let ,
,
be independently chosen vertices lying in the square with coordinates
,
,
, and
. The probability that the centroid of triangle
lies in the first quadrant is
for relatively prime positive integers
and
Find
Solution
Let have coordinates
,
have coordinates
,
and
have coordinates
.
Note that all these coordinates are uniformly distributed between
and
.
Thus, we want to find the probability that
and
both hold, which are independent events.
If , then
.
Thus, there exists a bijection between when
and when
.
(The case of
occurs with probability
).
so the probability is
for the chance
.
Thus, the answer is thus