2022 AIME I Problems/Problem 15
Contents
Problem
Let
and
be positive real numbers satisfying the system of equations:
Then
can be written as
where
and
are relatively prime positive integers. Find
Solution 1 (geometric interpretation)
First, let define a triangle with side lengths ,
, and
, with altitude from
's equal to
.
, the left side of one equation in the problem.
Let be angle opposite the side with length
. Then the altitude has length
and thus
, so
and the side length
is equal to
.
We can symmetrically apply this to the two other equations/triangles.
By law of sines, we have , with
as the circumradius, same for all 3 triangles.
The circumcircle's central angle to a side is
l=1, \sqrt{2}
\sqrt{3}
, and
, respectively.
This means that by half angle arcs, we see that we have in some order, ,
, and
(not necessarily this order, but here it does not matter due to symmetry), satisfying that
,
, and
. Solving, we get
,
, and
.
We notice that
- kevinmathz
Solution 2 (pure algebraic trig, easy to follow)
(This eventually whittles down to the same concept as Solution 1)
Note that in each equation in this system, it is possible to factor ,
, or
from each term (on the left sides), since each of
,
, and
are positive real numbers. After factoring out accordingly from each terms one of
,
, or
, the system should look like this:
This should give off tons of trigonometry vibes. To make the connection clear,
,
, and
is a helpful substitution:
From each equation
can be factored out, and when every equation is divided by 2, we get:
which simplifies to (using the Pythagorean identity
):
which further simplifies to (using sine addition formula
):
Without loss of generality, taking the inverse sine of each equation yields a simple system:
giving solutions
,
,
. Since these unknowns are directly related to our original unknowns, there are consequent solutions for those:
,
, and
. When plugging into the expression
, noting that
helps to simplify this expression into:
Now, all the cosines in here are fairly standard:
,
,
and
. With some final calculations:
This is our answer in simplest form
, so
~Oxymoronic15
solution 3
Let , rewrite those equations
;
and solve for
Square both sides and simplify, to get three equations:
Square both sides again, and simplify to get three equations:
Subtract first and third equation, getting ,
Put it in first equation, getting ,
Since ,
and so the final answer is
~bluesoul
Solution 4
Denote ,
,
.
Hence, the system of equations given in the problem can be written as
Each equation above takes the following form:
Now, we simplify this equation by removing radicals.
Denote and
.
Hence, the equation above implies
Hence, .
Hence,
.
Because and
, we get
.
Plugging this into the equation
and simplifying it, we get
Therefore, the system of equations above can be simplified as
Denote .
The system of equations above can be equivalently written as
Taking , we get
Thus, we have either or
.
:
.
Equation (2') implies .
Plugging and
into Equation (2), we get contradiction. Therefore, this case is infeasible.
:
.
Plugging this condition into (1') to substitute , we get
Taking , we get
Taking (4) + (5), we get
Hence, .
Therefore,
Therefore, the answer is .
\end{solution}
~Steven Chen (www.professorchenedu.com)
Solution 5
Let ,
, and
. Then,
Notice that ,
, and
. Let
,
, and
where
,
, and
are real. Substituting into
,
, and
yields
Thus,
so
. Hence,
so
, for a final answer of
.
Remark
The motivation for the trig substitution is that if , then
, and when making the substitution in each equation of the initial set of equations, we obtain a new equation in the form of the sine addition formula.
~ Leo.Euler
Solution 6 (Geometric)
In given equations, so we define some points:
Notice, that
and each points lies in the first quadrant.
We use given equations and get some scalar products:
So
Points and
are simmetric with respect to
Case 1
Case 2
vladimir.shelomovskii@gmail.com, vvsss
Video Solution
~Math Gold Medalist
Video Solution
https://www.youtube.com/watch?v=ihKUZ5itcdA
~Steven Chen (www.professorchenedu.com)
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Problem | |
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