2021 AIME I Problems/Problem 7
Problem
Find the number of pairs of positive integers with
such that there exists a real number
satisfying
Solution
The maximum value of is
, which is achieved at
for some integer
. This is left as an exercise to the reader.
This implies that , and that
and
, for integers
.
Taking their ratio, we have
It remains to find all
that satisfy this equation.
If , then
. This corresponds to choosing two elements from the set
. There are
ways to do so.
If , by multiplying
and
by the same constant
, we have that
. This means
. This corresponds to choosing two numbers from the set
. There are
ways here.
Finally, if , note that
must be an integer. This means that
belong to the set
, or
(why?). Taking casework on
, we get the sets
. Some sets have been omitted; this is because they were counted in the other cases already. This sums to
.
In total, there are pairs of
.
This solution was brought to you by ~Leonard_my_dude~.
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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