2022 AIME I Problems/Problem 4
Problem
Let and
where
Find the number of ordered pairs
of positive integers not exceeding
that satisfy the equation
Solution
We rewrite and
in polar form:
The equation
becomes
for some integer
Since and
we conclude that
Note that the values for
and the values for
have one-to-one correspondence.
We apply casework to the values for
There are values for
so there are
values for
It follows that
so there are
values for
There are ordered pairs
in this case.
There are values for
so there are
values for
It follows that
so there are
values for
There are ordered pairs
in this case.
There are values for
so there are
values for
It follows that
so there are
values for
There are ordered pairs
in this case.
Together, the answer is
~MRENTHUSIASM ~bluesoul
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=XiEaCq5jf5s
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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