2020 AIME II Problems/Problem 1
Contents
Problem
Find the number of ordered pairs of positive integers such that
.
Solution
First, we find the prime factorization of , which is
. The equation
tells us that we want to select a perfect square factor of
,
. The
might throw you off here, but it's actually kind of irrelevant because once
is selected, the remaining factor will already be assigned as
. There are
ways to select a perfect square factor of
, thus our answer is
.
~superagh
Solution 2 (Official MAA)
Because , if
, there must be nonnegative integers
,
,
, and
such that
and
. Then
and
The first equation has
solutions corresponding to
, and the second equation has
solutions corresponding to
. Therefore there are a total of
ordered pairs
such that
.
Video Solution
https://youtu.be/zfChnbMGLVQ?t=4612
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=x0QznvXcwHY
~IceMatrix
Video Solution 2
~avn
Purple Comet Math Meet April 2020
Notice, that this was the exact same problem (with different wording of course) as Purple Comet HS problem 3 and remembering the answer, put .
https://purplecomet.org/views/data/2020HSSolutions.pdf
~Lopkiloinm
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.