Difference between revisions of "2021 AMC 12A Problems/Problem 18"
Pi is 3.14 (talk | contribs) (→Video Solution by Punxsutawney Phil) |
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==Video Solution by Punxsutawney Phil== | ==Video Solution by Punxsutawney Phil== | ||
https://youtu.be/8gGcj95rlWY | https://youtu.be/8gGcj95rlWY | ||
+ | |||
+ | == Video Solution by pi_is_3.14 (Using Functions and manipulations) == | ||
+ | https://youtu.be/aGv99CLzguE | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==See also== | ==See also== | ||
{{AMC12 box|year=2021|ab=A|num-b=17|num-a=19}} | {{AMC12 box|year=2021|ab=A|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:27, 11 February 2021
Contents
Problem
Let be a function defined on the set of positive rational numbers with the property that
for all positive rational numbers
and
. Furthermore, suppose that
also has the property that
for every prime number
. For which of the following numbers
is
?
Solution 1
Looking through the solutions we can see that can be expressed as
so using the prime numbers to piece together what we have we can get
, so
or
.
-Lemonie
Solution 2
We know that . Adding
to both sides, we get
Also
In
we have
.
In we have
.
In we have
.
In we have
.
In we have
.
Thus, our answer is
~JHawk0224
Video Solution by Hawk Math
https://www.youtube.com/watch?v=dvlTA8Ncp58
Video Solution by Punxsutawney Phil
Video Solution by pi_is_3.14 (Using Functions and manipulations)
~ pi_is_3.14
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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