Difference between revisions of "2010 AIME I Problems/Problem 9"
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==Video Solution== | ==Video Solution== | ||
− | https://youtu.be/LXct4j_rYfw | + | https://youtu.be/LXct4j_rYfw (Video unavailable as of 20240829) |
~Shreyas S | ~Shreyas S |
Latest revision as of 08:57, 29 August 2024
Contents
Problem
Let be the real solution of the system of equations
,
,
. The greatest possible value of
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Solution
Solution 1
Add the three equations to get . Now, let
.
,
and
, so
. Now cube both sides; the
terms cancel out. Solve the remaining quadratic to get
. To maximize
choose
and so the sum is
giving
.
Solution 2
This is almost the same as solution 1. Note . Next, let
. Note that
and
, so we have
. Move 28 over, divide both sides by 3, then cube to get
. The
terms cancel out, so solve the quadratic to get
. We maximize
by choosing
, which gives us
. Thus, our answer is
.
Solution 3
We have that ,
, and
. Multiplying the three equations, and letting
, we have that
, and reducing, that
, which has solutions
. Adding the three equations and testing both solutions, we find the answer of
, so the desired quantity is
.
Video Solution by OmegaLearn
https://youtu.be/SpSuqWY01SE?t=1293
~ pi_is_3.14
Remark
It is tempting to add the equations and then use the well-known factorization . Unfortunately such a factorization is just a red herring: it doesn't give much information on
.
Another Remark
The real problem with adding the equations is that are real numbers based on the problem, but the adding trick only works when
are integers.
Video Solution
https://youtu.be/LXct4j_rYfw (Video unavailable as of 20240829)
~Shreyas S
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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.