Difference between revisions of "2020 AIME II Problems/Problem 1"
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<cmath>2b+d = 20</cmath> | <cmath>2b+d = 20</cmath> | ||
− | The first equation has <math>21</math> solutions corresponding to <math>a = 0,1,2,\dots,20</math>, and the second equation has <math>11</math> solutions corresponding to <math>b = 0,1,2,\dots,10</math>. Therefore there are a total of <math>21\cdot11 = 231</math> ordered pairs <math>(m,n)</math> such that <math>m^2n = 20^{20}</math>. | + | The first equation has <math>21</math> solutions corresponding to <math>a = 0,1,2,\dots,20</math>, and the second equation has <math>11</math> solutions corresponding to <math>b = 0,1,2,\dots,10</math>. Therefore there are a total of <math>21\cdot11 = \boxed{231}</math> ordered pairs <math>(m,n)</math> such that <math>m^2n = 20^{20}</math>. |
− | + | == Video Solution by OmegaLearn== | |
− | == Video Solution == | ||
https://youtu.be/zfChnbMGLVQ?t=4612 | https://youtu.be/zfChnbMGLVQ?t=4612 | ||
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~Lopkiloinm | ~Lopkiloinm | ||
+ | |||
+ | ==Video Solution by WhyMath== | ||
+ | https://youtu.be/Gs27CPxRiTA | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== |
Latest revision as of 13:59, 25 February 2024
Contents
Problem
Find the number of ordered pairs of positive integers such that
.
Solution
In this problem, we want to find the number of ordered pairs such that
. Let
. Therefore, we want two numbers,
and
, such that their product is
and
is a perfect square. Note that there is exactly one valid
for a unique
, which is
. This reduces the problem to finding the number of unique perfect square factors of
.
Therefore, the answer is
~superagh
~TheBeast5520
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 by OmegaLearn
https://youtu.be/zfChnbMGLVQ?t=4612
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=VA1lReSkGXU
~ North America Math Contest Go Go Go
Video Solution
https://www.youtube.com/watch?v=x0QznvXcwHY
~IceMatrix
Video Solution
~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
Video Solution by WhyMath
~savannahsolver
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.