Difference between revisions of "2022 AIME I Problems/Problem 15"
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Then <math>\left[ (1-x)(1-y)(1-z) \right]^2</math> can be written as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | Then <math>\left[ (1-x)(1-y)(1-z) \right]^2</math> can be written as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
− | ==Solution 1 (easy to follow)== | + | ==Solution 1 (geometric interpretation)== |
+ | |||
+ | Favorite problem on the test. Extremely clean. (Solution close to that in post #2) | ||
+ | |||
+ | First, we note that we can let a triangle exist with side lengths <math>\sqrt{2x}</math>, <math>\sqrt{2z}</math>, and opposite altitude <math>\sqrt{xz}</math>. This shows that the third side, which is the nasty square-rooted sum, is going to have the length equal to the sum on the right - let this be <math>l</math> for symmetry purposes. So, we note that if the angle opposite the side with length <math>\sqrt{2x}</math> has a value of <math>\sin(\theta)</math>, then the altitude has length <math>\sqrt{2z} \cdot \sin(\theta) = \sqrt{xz}</math> and thus <math>\sin(\theta) = \sqrt{\frac{x}{2}}</math> so <math>x=2\sin^2(\theta)</math> and the triangle side with length <math>\sqrt{2x}</math> is equal to <math>2\sin(\theta)</math>. | ||
+ | |||
+ | We can symmetrically apply this to the two other triangles, and since by law of sines, we have <math>\frac{2\sin(\theta)}{\sin(\theta)} = 2R \to R=1</math> is the circumradius of that triangle. Hence. we calculate that with <math>l=1, \sqrt{2}</math>, and <math>\sqrt{3}</math>, the angles from the third side with respect to the circumcenter are <math>120^{\circ}, 90^{\circ}</math>, and <math>60^{\circ}</math>. This means that by half angle arcs, we see that we have in some order, <math>x=2\sin^2(\alpha)</math>, <math>x=2\sin^2(\beta)</math>, and <math>z=2\sin^2(\gamma)</math> (not necessarily this order, but here it does not matter due to symmetry), satisfying that <math>\alpha+\beta=180^{\circ}-\frac{120^{\circ}}{2}</math>, <math>\beta+\gamma=180^{\circ}-\frac{90^{\circ}}{2}</math>, and <math>\gamma+\alpha=180^{\circ}-\frac{60^{\circ}}{2}</math>. Solving, we get <math>\alpha=\frac{135^{\circ}}{2}</math>, <math>\beta=\frac{105^{\circ}}{2}</math>, and <math>\gamma=\frac{165^{\circ}}{2}</math>. | ||
+ | |||
+ | We notice that <cmath>[(1-x)(1-y)(1-z)]^2=[\sin(2\alpha)\sin(2\beta)\sin(2\gamma)]^2=[\sin(135^{\circ})\sin(105^{\circ})\sin(165^{\circ})]^2</cmath> <cmath>=\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{6}-\sqrt{2}}{4} \cdot \frac{\sqrt{6}+\sqrt{2}}{4}\right)^2 = \left(\frac{\sqrt{2}}{8}\right)^2=\frac{1}{32} \to \boxed{033}. \blacksquare</cmath> | ||
+ | |||
+ | ==Solution 2 (easy to follow)== | ||
Note that in each equation in this system, it is possible to factor <math>\sqrt{x}</math>, <math>\sqrt{y}</math>, or <math>\sqrt{z}</math> from each term (on the left sides), since each of <math>x</math>, <math>y</math>, and <math>z</math> are positive real numbers. After factoring out accordingly from each terms one of <math>\sqrt{x}</math>, <math>\sqrt{y}</math>, or <math>\sqrt{z}</math>, the system should look like this: | Note that in each equation in this system, it is possible to factor <math>\sqrt{x}</math>, <math>\sqrt{y}</math>, or <math>\sqrt{z}</math> from each term (on the left sides), since each of <math>x</math>, <math>y</math>, and <math>z</math> are positive real numbers. After factoring out accordingly from each terms one of <math>\sqrt{x}</math>, <math>\sqrt{y}</math>, or <math>\sqrt{z}</math>, the system should look like this: |
Revision as of 21:37, 17 February 2022
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)
Favorite problem on the test. Extremely clean. (Solution close to that in post #2)
First, we note that we can let a triangle exist with side lengths ,
, and opposite altitude
. This shows that the third side, which is the nasty square-rooted sum, is going to have the length equal to the sum on the right - let this be
for symmetry purposes. So, we note that if the angle opposite the side with length
has a value of
, then the altitude has length
and thus
so
and the triangle side with length
is equal to
.
We can symmetrically apply this to the two other triangles, and since by law of sines, we have is the circumradius of that triangle. Hence. we calculate that with
, and
, the angles from the third side with respect to the circumcenter are
, and
. 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
Solution 2 (easy to follow)
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, we can take the inverse sine of each equation. The resulting system is simple:
giving solutions
,
,
. Since these unknowns are directly related to our original unknowns, we also have solutions for those:
,
, and
. When plugging into the expression
(Still editing) - Oxymoronic15