2021 JMC 10 Problems/Problem 16
Problem
If and
are randomly chosen numbers between
and
, what is the probability that
(Recall that
denotes the greatest integer less than or equal to
)
Solution
We use an area-based approach along with number lines, where are total area is . We can stop worrying about what happens when at least one of
and
is an integer, because those cases are negligible. Below is the number line representing the value of
in the intervals. If seeing what happens on
is too complicated, plugging in
will suffice.
Below is the number line representing the value of in the intervals.
We seek the number of pairs of
and
intervals where
. Doing simple casework on the equal value, we have
cases, so our answer is
.