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Difference between revisions of "2022 AIME I Problems/Problem 15"
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==Problem==
 
==Problem==
 +
Let <math>x,</math> <math>y,</math> and <math>z</math> be positive real numbers satisfying the system of equations:
 +
<cmath>\begin{align*}
 +
\sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\
 +
\sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\
 +
\sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3.
 +
\end{align*}</cmath>
 +
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 (geometric interpretation)==
 +
 +
First, let define a triangle with side lengths <math>\sqrt{2x}</math>, <math>\sqrt{2z}</math>, and <math>l</math>, with altitude from <math>l</math>'s equal to <math>\sqrt{xz}</math>. <math>l = \sqrt{2x - xz} + \sqrt{2z - xz}</math>, the left side of one equation in the problem.
 +
 +
Let  <math>\theta</math> be angle opposite the side with length <math>\sqrt{2x}</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 side length <math>\sqrt{2x}</math> is equal to <math>2\sin(\theta)</math>.
 +
 +
We can symmetrically apply this to the two other equations/triangles.
 +
 +
By law of sines, we have <math>\frac{2\sin(\theta)}{\sin(\theta)} = 2R</math>, with <math>R=1</math> as the circumradius, same for all 3 triangles.
 +
The circumcircle's central angle to a side is <math>2 \arcsin(l/2)</math>, so the 3 triangles' <math>l=1, \sqrt{2}, \sqrt{3}</math>,  have angles <math>120^{\circ}, 90^{\circ}, 60^{\circ}</math>, respectively.
 +
 +
This means that by half angle arcs, we see that we have in some order, <math>x=2\sin^2(\alpha)</math>, <math>y=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>
 +
 +
- 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 <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:
 +
<cmath>\begin{align*}
 +
\sqrt{x}\cdot\sqrt{2-y} + \sqrt{y}\cdot\sqrt{2-x} &= 1 \\
 +
\sqrt{y}\cdot\sqrt{2-z} + \sqrt{z}\cdot\sqrt{2-y} &= \sqrt2 \\
 +
\sqrt{z}\cdot\sqrt{2-x} + \sqrt{x}\cdot\sqrt{2-z} &= \sqrt3.
 +
\end{align*}</cmath>
 +
 +
This should give off tons of trigonometry vibes. To make the connection clear, <math>x = 2\cos^2 \alpha</math>, <math>y = 2\cos^2 \beta</math>, and <math>z = 2\cos^2 \theta</math> is a helpful substitution:
 +
 +
<cmath>\begin{align*}
 +
\sqrt{2\cos^2 \alpha}\cdot\sqrt{2-2\cos^2 \beta} + \sqrt{2\cos^2 \beta}\cdot\sqrt{2-2\cos^2 \alpha} &= 1 \\
 +
\sqrt{2\cos^2 \beta}\cdot\sqrt{2-2\cos^2 \theta} + \sqrt{2\cos^2 \theta}\cdot\sqrt{2-2\cos^2 \beta} &= \sqrt2 \\
 +
\sqrt{2\cos^2 \theta}\cdot\sqrt{2-2\cos^2 \alpha} + \sqrt{2\cos^2 \alpha}\cdot\sqrt{2-2\cos^2 \theta} &= \sqrt3.
 +
\end{align*}</cmath>
 +
 +
From each equation <math>\sqrt{2}^2</math> can be factored out, and when every equation is divided by 2, we get:
 +
 +
<cmath>\begin{align*}
 +
\sqrt{\cos^2 \alpha}\cdot\sqrt{1-\cos^2 \beta} + \sqrt{\cos^2 \beta}\cdot\sqrt{1-\cos^2 \alpha} &= \frac{1}{2} \\
 +
\sqrt{\cos^2 \beta}\cdot\sqrt{1-\cos^2 \theta} + \sqrt{\cos^2 \theta}\cdot\sqrt{1-\cos^2 \beta} &= \frac{\sqrt2}{2} \\
 +
\sqrt{\cos^2 \theta}\cdot\sqrt{1-\cos^2 \alpha} + \sqrt{\cos^2 \alpha}\cdot\sqrt{1-\cos^2 \theta} &= \frac{\sqrt3}{2}.
 +
\end{align*}</cmath>
 +
 +
which simplifies to (using the Pythagorean identity <math>\sin^2 \phi + \cos^2 \phi = 1 \; \forall \; \phi \in \mathbb{C} </math>):
 +
 +
<cmath>\begin{align*}
 +
\cos \alpha\cdot\sin \beta + \cos \beta\cdot\sin \alpha &= \frac{1}{2} \\
 +
\cos \beta\cdot\sin \theta + \cos \theta\cdot\sin \beta &= \frac{\sqrt2}{2} \\
 +
\cos \theta\cdot\sin \alpha + \cos \alpha\cdot\sin \theta &= \frac{\sqrt3}{2}.
 +
\end{align*}</cmath>
 +
 +
which further simplifies to (using sine addition formula <math>\sin(a + b) = \sin a \cos b + \cos a \sin b</math>):
 +
 +
<cmath>\begin{align*}
 +
\sin(\alpha + \beta) &= \frac{1}{2} \\
 +
\sin(\beta + \theta) &= \frac{\sqrt2}{2} \\
 +
\sin(\alpha + \theta) &= \frac{\sqrt3}{2}.
 +
\end{align*}</cmath>
 +
 +
Taking the inverse sine (<math>0\leq\theta\frac{\pi}{2}</math>) of each equation yields a simple system:
 +
 +
<cmath>\begin{align*}
 +
\alpha + \beta &= \frac{\pi}{6} \\
 +
\beta + \theta &= \frac{\pi}{4} \\
 +
\alpha + \theta &= \frac{\pi}{3}
 +
\end{align*}</cmath>
 +
 +
giving solutions:
 +
 +
<cmath>\begin{align*}
 +
\alpha &= \frac{\pi}{8} \\
 +
\beta &= \frac{\pi}{24} \\
 +
\theta &= \frac{5\pi}{24}
 +
\end{align*}</cmath>
 +
 +
Since these unknowns are directly related to our original unknowns, there are consequent solutions for those:
 +
 +
 +
<cmath>\begin{align*}
 +
x &= 2\cos^2\left(\frac{\pi}{8}\right) \\
 +
y &= 2\cos^2\left(\frac{\pi}{24}\right) \\
 +
z &= 2\cos^2\left(\frac{5\pi}{24}\right)
 +
\end{align*}</cmath>
 +
 +
When plugging into the expression <math>\left[ (1-x)(1-y)(1-z) \right]^2</math>, noting that <math>-\cos 2\phi = 1 - 2\cos^2 \phi\; \forall \; \phi \in \mathbb{C}</math> helps to simplify this expression into:
 +
 +
 +
<cmath>\begin{align*}
 +
\left[ (-1)^3\left(\cos \left(2\cdot\frac{\pi}{8}\right)\cos \left(2\cdot\frac{\pi}{24}\right)\cos \left(2\cdot\frac{5\pi}{24}\right)\right)\right]^2 \\
 +
= \left[ (-1)\left(\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{5\pi}{12}\right)\right)\right]^2
 +
\end{align*}</cmath> 
 +
 +
Now, all the cosines in here are fairly standard:
 +
 +
<cmath>\begin{align*}
 +
\cos \frac{\pi}{4} &= \frac{\sqrt{2}}{2} \\
 +
\cos \frac{\pi}{12} &=\frac{\sqrt{6} + \sqrt{2}}{4} & (= \cos{\frac{\frac{\pi}{6}}{2}} ) \\
 +
\cos \frac{5\pi}{12} &= \frac{\sqrt{6} - \sqrt{2}}{4} & (=  \cos\left({\frac{\pi}{6} + \frac{\pi}{4}} \right) )
 +
\end{align*}</cmath>
 +
 +
With some final calculations:
 +
 +
 +
<cmath>\begin{align*}
 +
&(-1)^2\left(\frac{\sqrt{2}}{2}\right)^2\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)^2\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2 \\
 +
=&
 +
\left(\frac{1}{2}\right)
 +
\left(\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)\right)^2 \\
 +
=&\frac{1}{2} \frac{4^2}{16^2} = \frac{1}{32}
 +
\end{align*}</cmath>
 +
 +
This is our answer in simplest form <math>\frac{m}{n}</math>, so <math>m + n = 1 + 32 = \boxed{033}</math>.
 +
 +
~Oxymoronic15
 +
 +
==Solution 3 (substitution)==
 +
Let <math>1-x=a;1-y=b;1-z=c</math>, rewrite those equations
 +
 +
<math>\sqrt{(1-a)(1+b)}+\sqrt{(1+a)(1-b)}=1</math>;
 +
 +
<math>\sqrt{(1-b)(1+c)}+\sqrt{(1+b)(1-c)}=\sqrt{2}</math>
 +
 +
<math>\sqrt{(1-a)(1+c)}+\sqrt{(1-c)(1+a)}=\sqrt{3}</math>
 +
 +
and solve for <math>m/n = (abc)^2 = a^2b^2c^2</math>
 +
 +
Square both sides and simplify, to get three equations:
 +
 +
<math>2ab-1=2\sqrt{(1-a^2)(1-b^2)}</math>
 +
 +
<math>2bc~ ~ ~ ~  ~ ~=2\sqrt{(1-b^2)(1-c^2)}</math>
 +
 +
<math>2ac+1=2\sqrt{(1-c^2)(1-a^2)}</math>
 +
 +
Square both sides again, and simplify to get three equations:
 +
 +
<math>a^2+b^2-ab=\frac{3}{4}</math>
 +
 +
<math>b^2+c^2~ ~ ~ ~ ~ ~=1</math>
 +
 +
<math>a^2+c^2+ac=\frac{3}{4}</math>
 +
 +
Subtract first and third equation, getting <math>(b+c)(b-c)=a(b+c)</math>, <math>a=b-c</math>
 +
 +
Put it in first equation, getting <math>b^2-2bc+c^2+b^2-b(b-c)=b^2+c^2-bc=\frac{3}{4}</math>, <math>bc=\frac{1}{4}</math>
 +
 +
Since <math>a^2=b^2+c^2-2bc=\frac{1}{2}</math>, <math>m/n = a^2b^2c^2 = a^2(bc)^2 = \frac{1}{2}\left(\frac{1}{4}\right)^2=\frac{1}{32}</math>  and so the final answer is <math>\boxed{033}</math>
 +
 +
~bluesoul
 +
 +
==Solution 4==
 +
 +
Denote <math>u = 1 - x</math>, <math>v = 1 - y</math>, <math>w = 1 - z</math>.
 +
Hence, the system of equations given in the problem can be written as
 +
<cmath>
 +
\begin{align*}
 +
\sqrt{(1-u)(1+v)} + \sqrt{(1+u)(1-v)} & = 1 \hspace{1cm} (1) \\
 +
\sqrt{(1-v)(1+w)} + \sqrt{(1+v)(1-w)} & = \sqrt{2} \hspace{1cm} (2) \\
 +
\sqrt{(1-w)(1+u)} + \sqrt{(1+w)(1-u)} & = \sqrt{3} . \hspace{1cm} (3)
 +
\end{align*}
 +
</cmath>
 +
 +
Each equation above takes the following form:
 +
<cmath>
 +
\[
 +
\sqrt{(1-a)(1+b)} + \sqrt{(1+a)(1-b)} = k .
 +
\]
 +
</cmath>
 +
 +
Now, we simplify this equation by removing radicals.
 +
 +
Denote <math>p = \sqrt{(1-a)(1+b)}</math> and <math>q = \sqrt{(1+a)(1-b)}</math>.
 +
 +
Hence, the equation above implies
 +
<cmath>
 +
\[
 +
\left\{
 +
\begin{array}{l}
 +
p + q = k \\
 +
p^2 = (1-a)(1+b) \\
 +
q^2 = (1+a)(1-b)
 +
\end{array}
 +
\right..
 +
\]
 +
</cmath>
 +
 +
Hence, <math>q^2 - p^2 = (1+a)(1-b) - (1-a)(1+b) = 2 (a-b)</math>.
 +
Hence, <math>q - p = \frac{q^2 - p^2}{p+q} = \frac{2}{k} (a-b)</math>.
 +
 +
Because <math>p + q = k</math> and <math>q - p = \frac{2}{k} (a-b)</math>, we get <math>q = \frac{a-b}{k} + \frac{k}{2}</math>.
 +
Plugging this into the equation <math>q^2 = (1+a)(1-b)</math> and simplifying it, we get
 +
<cmath>
 +
\[
 +
a^2 + \left( k^2 - 2 \right) ab + b^2 = k^2 - \frac{k^4}{4} .
 +
\]
 +
</cmath>
 +
 +
Therefore, the system of equations above can be simplified as
 +
<cmath>
 +
\begin{align*}
 +
u^2 - uv + v^2 & = \frac{3}{4} \\
 +
v^2 + w^2 & = 1 \\
 +
w^2 + wu + u^2 & = \frac{3}{4}  .
 +
\end{align*}
 +
</cmath>
 +
 +
Denote <math>w' = - w</math>.
 +
The system of equations above can be equivalently written as
 +
<cmath>
 +
\begin{align*}
 +
u^2 - uv + v^2 & = \frac{3}{4} \hspace{1cm} (1') \\
 +
v^2 + w'^2 & = 1 \hspace{1cm} (2') \\
 +
w'^2 - w'u + u^2 & = \frac{3}{4} \hspace{1cm} (3') .
 +
\end{align*}
 +
</cmath>
 +
 +
Taking <math>(1') - (3')</math>, we get
 +
<cmath>
 +
\[
 +
(v - w') (v + w' - u) = 0 .
 +
\]
 +
</cmath>
 +
 +
Thus, we have either <math>v - w' = 0</math> or <math>v + w' - u = 0</math>.
 +
 +
<math>\textbf{Case 1}</math>: <math>v - w' = 0</math>.
 +
 +
Equation (2') implies <math>v = w' = \pm \frac{1}{\sqrt{2}}</math>.
 +
 +
Plugging <math>v</math> and <math>w'</math> into Equation (2), we get contradiction. Therefore, this case is infeasible.
 +
 +
<math>\textbf{Case 2}</math>: <math>v + w' - u = 0</math>.
 +
 +
Plugging this condition into (1') to substitute <math>u</math>, we get
 +
<cmath>
 +
\[
 +
v^2 + v w' + w'^2 = \frac{3}{4} \hspace{1cm} (4) .
 +
\]
 +
</cmath>
 +
 +
Taking <math>(4) - (2')</math>, we get
 +
<cmath>
 +
\[
 +
v w' = - \frac{1}{4} . \hspace{1cm} (5) .
 +
\]
 +
</cmath>
 +
 +
Taking (4) + (5), we get
 +
<cmath>
 +
\[
 +
\left( v + w' \right)^2 = \frac{1}{2} .
 +
\]
 +
</cmath>
 +
 +
Hence, <math>u^2 = \left( v + w' \right)^2 = \frac{1}{2}</math>.
 +
 +
Therefore,
 +
<cmath>
 +
\begin{align*}
 +
\left[ (1-x)(1-y)(1-z) \right]^2
 +
& = u^2 (vw)^2 \\
 +
& = u^2 (vw')^2 \\
 +
& = \frac{1}{2} \left( - \frac{1}{4} \right)^2 \\
 +
& = \frac{1}{32} .
 +
\end{align*}
 +
</cmath>
 +
 +
Therefore, the answer is <math>1 + 32 = \boxed{\textbf{(033) }}</math>.
 +
 +
~Steven Chen (www.professorchenedu.com)
 +
 +
bu-bye
 +
<cmath>\begin{align*}
 +
\sin(\alpha + \beta) &= 1/2 \\
 +
\sin(\alpha + \gamma) &= \sqrt2/2 \\
 +
\sin(\beta + \gamma) &= \sqrt3/2.
 +
\end{align*}</cmath>
 +
Thus,
 +
<cmath>\begin{align*}
 +
\alpha + \beta &= 30^{\circ} \\
 +
\alpha + \gamma &= 45^{\circ} \\
 +
\beta + \gamma &= 60^{\circ},
 +
\end{align*}</cmath>
 +
so <math>(\alpha, \beta, \gamma) = (15/2^{\circ}, 45/2^{\circ}, 75/2^{\circ})</math>. Hence,
 +
 +
<cmath> abc = (1-2\sin^2(\alpha))(1-2\sin^2(\beta))(1-2\sin^2(\gamma))=\cos(15^{\circ})\cos(45^{\circ})\cos(75^{\circ})=\frac{\sqrt{2}}{8}, </cmath>
 +
so <math>(abc)^2=(\sqrt{2}/8)^2=\frac{1}{32}</math>, for a final answer of <math>\boxed{033}</math>.
 +
 +
<u><b>Remark</b></u>
 +
 +
The motivation for the trig substitution is that if <math>\sin^2(\alpha)=(1-a)/2</math>, then <math>\cos^2(\alpha)=(1+a)/2</math>, 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)==
 +
[[File:2022 AIME I 15.png|400px|right]]
 +
In given equations, <math>0 \leq x,y,z \leq 2,</math> so we define some points:
 +
<cmath>\bar {O} = (0, 0), \bar {A} = (1, 0),
 +
\bar{M} = \left(\frac {1}{\sqrt{2}},\frac {1}{\sqrt{2}}\right),</cmath>
 +
<cmath>\bar {X} = \left(\sqrt {\frac {x}{2}}, \sqrt{1 – \frac{x}{2}}\right),
 +
\bar {Y'} = \left(\sqrt {\frac {y}{2}}, \sqrt{1 – \frac{y}{2}}\right),</cmath>
 +
<cmath>\bar {Y} = \left(\sqrt {1 – \frac{y}{2}},\sqrt{\frac {y}{2}}\right),
 +
\bar {Z} = \left(\sqrt {1 – \frac{z}{2}},\sqrt{\frac {z}{2}}\right).</cmath>
 +
Notice, that <cmath>\mid \vec {AO} \mid = \mid \vec {MO} \mid = \mid \vec {XO} \mid =\mid \vec {YO} \mid = \mid \vec {Y'O} \mid =\mid \vec {ZO} \mid = 1</cmath> and each points lies in the first quadrant.
 +
 +
We use given equations and get some scalar products:
 +
<cmath>(\vec {XO} \cdot \vec {YO}) = \frac {1}{2} = \cos \angle XOY \implies  \angle XOY = 60 ^\circ,</cmath>
 +
<cmath>(\vec {XO} \cdot \vec {ZO}) = \frac {\sqrt{3}}{2} = \cos \angle XOZ \implies  \angle XOZ = 30^\circ,</cmath> 
 +
<cmath>(\vec {Y'O} \cdot \vec {ZO}) = \frac {1}{\sqrt{2}} = \cos \angle Y'OZ \implies  \angle Y'OZ = 45^\circ.</cmath>
 +
So <math> \angle YOZ =  \angle XOY – \angle XOZ =  60 ^\circ – 30 ^\circ = 30 ^\circ, 
 +
\angle Y'OY =  \angle Y'OZ +  \angle YOZ = 45^\circ + 30 ^\circ = 75^\circ.</math>
 +
 +
Points <math>Y</math> and <math>Y'</math> are symmetric with respect to <math>OM.</math>
 +
 +
<i><b>Case 1</b></i>
 +
<cmath>\angle YOA = \frac{90^\circ – 75^\circ}{2} = 7.5^\circ, \angle ZOA = 30^\circ + 7.5^\circ = 37.5^\circ, \angle XOA = 60^\circ + 7.5^\circ = 67.5^\circ .</cmath>
 +
<cmath>1 – x = \left(\sqrt{1 – \frac{x}{2}} \right)^2– \left(\sqrt{\frac {x}{2}}\right)^2  = \sin^2 \angle XOA – \cos^2 \angle XOA = –\cos 2 \angle XOA = –\cos 135^\circ,</cmath>
 +
<cmath>1 – y = \cos 15^\circ, 1 – z = \cos 75^\circ \implies  \left[ (1–x)(1–y)(1–z) \right]^2 = \left[ \sin 45^\circ \cdot \cos 15^\circ \cdot \sin 15^\circ \right]^2 =</cmath>
 +
<cmath>=\left[ \frac {\sin 45^\circ \cdot \sin 30^\circ}{2} \right]^2  = \frac {1}{32} \implies \boxed{\textbf{033}}.</cmath>
 +
<i><b>Case 2</b></i>
 +
 +
<cmath>\angle Y_1 OA = \frac{90^\circ + 75^\circ}{2} = 82.5^\circ, \angle Z_1 OA = 82.5^\circ – 30^\circ = 52.5^\circ,
 +
\angle X_1 OA = 82.5^\circ – 60^\circ = 22.5^\circ \implies \boxed{\textbf{033}}.</cmath>
 +
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
 +
==Video Solution==
 +
https://youtu.be/i6kDMbav2sk
 +
 +
~Math Gold Medalist
 +
 +
 +
==Video Solution==
 +
 +
https://youtu.be/aa_VY4e4OOM?si=1lHSwY3v7RICoEpk
 +
 +
~MathProblemSolvingSkills.com
 +
 +
 +
 +
==Video Solution==
 +
 +
https://www.youtube.com/watch?v=ihKUZ5itcdA
 +
 +
~Steven Chen (www.professorchenedu.com)
 +
 +
==See Also==
 +
{{AIME box|year=2022|n=I|num-b=14|after=Last Problem}}
 +
 +
{{MAA Notice}}

Latest revision as of 00:25, 1 February 2024

Problem

Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations: \begin{align*} \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\ \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\ \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. \end{align*} Then $\left[ (1-x)(1-y)(1-z) \right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution 1 (geometric interpretation)

First, let define a triangle with side lengths $\sqrt{2x}$, $\sqrt{2z}$, and $l$, with altitude from $l$'s equal to $\sqrt{xz}$. $l = \sqrt{2x - xz} + \sqrt{2z - xz}$, the left side of one equation in the problem.

Let $\theta$ be angle opposite the side with length $\sqrt{2x}$. Then the altitude has length $\sqrt{2z} \cdot \sin(\theta) = \sqrt{xz}$ and thus $\sin(\theta) = \sqrt{\frac{x}{2}}$, so $x=2\sin^2(\theta)$ and the side length $\sqrt{2x}$ is equal to $2\sin(\theta)$.

We can symmetrically apply this to the two other equations/triangles.

By law of sines, we have $\frac{2\sin(\theta)}{\sin(\theta)} = 2R$, with $R=1$ as the circumradius, same for all 3 triangles. The circumcircle's central angle to a side is $2 \arcsin(l/2)$, so the 3 triangles' $l=1, \sqrt{2}, \sqrt{3}$, have angles $120^{\circ}, 90^{\circ}, 60^{\circ}$, respectively.

This means that by half angle arcs, we see that we have in some order, $x=2\sin^2(\alpha)$, $y=2\sin^2(\beta)$, and $z=2\sin^2(\gamma)$ (not necessarily this order, but here it does not matter due to symmetry), satisfying that $\alpha+\beta=180^{\circ}-\frac{120^{\circ}}{2}$, $\beta+\gamma=180^{\circ}-\frac{90^{\circ}}{2}$, and $\gamma+\alpha=180^{\circ}-\frac{60^{\circ}}{2}$. Solving, we get $\alpha=\frac{135^{\circ}}{2}$, $\beta=\frac{105^{\circ}}{2}$, and $\gamma=\frac{165^{\circ}}{2}$.

We notice that \[[(1-x)(1-y)(1-z)]^2=[\sin(2\alpha)\sin(2\beta)\sin(2\gamma)]^2=[\sin(135^{\circ})\sin(105^{\circ})\sin(165^{\circ})]^2\] \[=\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{6}-\sqrt{2}}{4} \cdot \frac{\sqrt{6}+\sqrt{2}}{4}\right)^2 = \left(\frac{\sqrt{2}}{8}\right)^2=\frac{1}{32} \to \boxed{033}. \blacksquare\]

- 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 $\sqrt{x}$, $\sqrt{y}$, or $\sqrt{z}$ from each term (on the left sides), since each of $x$, $y$, and $z$ are positive real numbers. After factoring out accordingly from each terms one of $\sqrt{x}$, $\sqrt{y}$, or $\sqrt{z}$, the system should look like this: \begin{align*} \sqrt{x}\cdot\sqrt{2-y} + \sqrt{y}\cdot\sqrt{2-x} &= 1 \\ \sqrt{y}\cdot\sqrt{2-z} + \sqrt{z}\cdot\sqrt{2-y} &= \sqrt2 \\ \sqrt{z}\cdot\sqrt{2-x} + \sqrt{x}\cdot\sqrt{2-z} &= \sqrt3. \end{align*}

This should give off tons of trigonometry vibes. To make the connection clear, $x = 2\cos^2 \alpha$, $y = 2\cos^2 \beta$, and $z = 2\cos^2 \theta$ is a helpful substitution:

\begin{align*} \sqrt{2\cos^2 \alpha}\cdot\sqrt{2-2\cos^2 \beta} + \sqrt{2\cos^2 \beta}\cdot\sqrt{2-2\cos^2 \alpha} &= 1 \\ \sqrt{2\cos^2 \beta}\cdot\sqrt{2-2\cos^2 \theta} + \sqrt{2\cos^2 \theta}\cdot\sqrt{2-2\cos^2 \beta} &= \sqrt2 \\ \sqrt{2\cos^2 \theta}\cdot\sqrt{2-2\cos^2 \alpha} + \sqrt{2\cos^2 \alpha}\cdot\sqrt{2-2\cos^2 \theta} &= \sqrt3. \end{align*}

From each equation $\sqrt{2}^2$ can be factored out, and when every equation is divided by 2, we get:

\begin{align*} \sqrt{\cos^2 \alpha}\cdot\sqrt{1-\cos^2 \beta} + \sqrt{\cos^2 \beta}\cdot\sqrt{1-\cos^2 \alpha} &= \frac{1}{2} \\ \sqrt{\cos^2 \beta}\cdot\sqrt{1-\cos^2 \theta} + \sqrt{\cos^2 \theta}\cdot\sqrt{1-\cos^2 \beta} &= \frac{\sqrt2}{2} \\ \sqrt{\cos^2 \theta}\cdot\sqrt{1-\cos^2 \alpha} + \sqrt{\cos^2 \alpha}\cdot\sqrt{1-\cos^2 \theta} &= \frac{\sqrt3}{2}. \end{align*}

which simplifies to (using the Pythagorean identity $\sin^2 \phi + \cos^2 \phi = 1 \; \forall \; \phi \in \mathbb{C}$):

\begin{align*} \cos \alpha\cdot\sin \beta + \cos \beta\cdot\sin \alpha &= \frac{1}{2} \\ \cos \beta\cdot\sin \theta + \cos \theta\cdot\sin \beta &= \frac{\sqrt2}{2} \\ \cos \theta\cdot\sin \alpha + \cos \alpha\cdot\sin \theta &= \frac{\sqrt3}{2}. \end{align*}

which further simplifies to (using sine addition formula $\sin(a + b) = \sin a \cos b + \cos a \sin b$):

\begin{align*} \sin(\alpha + \beta) &= \frac{1}{2} \\ \sin(\beta + \theta) &= \frac{\sqrt2}{2} \\ \sin(\alpha + \theta) &= \frac{\sqrt3}{2}. \end{align*}

Taking the inverse sine ($0\leq\theta\frac{\pi}{2}$) of each equation yields a simple system:

\begin{align*} \alpha + \beta &= \frac{\pi}{6} \\ \beta + \theta &= \frac{\pi}{4} \\ \alpha + \theta &= \frac{\pi}{3} \end{align*}

giving solutions:

\begin{align*} \alpha &= \frac{\pi}{8} \\ \beta &= \frac{\pi}{24} \\ \theta &= \frac{5\pi}{24} \end{align*}

Since these unknowns are directly related to our original unknowns, there are consequent solutions for those:


\begin{align*} x &= 2\cos^2\left(\frac{\pi}{8}\right) \\ y &= 2\cos^2\left(\frac{\pi}{24}\right) \\ z &= 2\cos^2\left(\frac{5\pi}{24}\right) \end{align*}

When plugging into the expression $\left[ (1-x)(1-y)(1-z) \right]^2$, noting that $-\cos 2\phi = 1 - 2\cos^2 \phi\; \forall \; \phi \in \mathbb{C}$ helps to simplify this expression into:


\begin{align*} \left[ (-1)^3\left(\cos \left(2\cdot\frac{\pi}{8}\right)\cos \left(2\cdot\frac{\pi}{24}\right)\cos \left(2\cdot\frac{5\pi}{24}\right)\right)\right]^2 \\ = \left[ (-1)\left(\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{5\pi}{12}\right)\right)\right]^2 \end{align*}

Now, all the cosines in here are fairly standard:

\begin{align*} \cos \frac{\pi}{4} &= \frac{\sqrt{2}}{2} \\ \cos \frac{\pi}{12} &=\frac{\sqrt{6} + \sqrt{2}}{4} & (= \cos{\frac{\frac{\pi}{6}}{2}} ) \\ \cos \frac{5\pi}{12} &= \frac{\sqrt{6} - \sqrt{2}}{4} & (= \cos\left({\frac{\pi}{6} + \frac{\pi}{4}} \right) ) \end{align*}

With some final calculations:


\begin{align*} &(-1)^2\left(\frac{\sqrt{2}}{2}\right)^2\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)^2\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2 \\ =& \left(\frac{1}{2}\right) \left(\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)\right)^2 \\ =&\frac{1}{2} \frac{4^2}{16^2} = \frac{1}{32} \end{align*}

This is our answer in simplest form $\frac{m}{n}$, so $m + n = 1 + 32 = \boxed{033}$.

~Oxymoronic15

Solution 3 (substitution)

Let $1-x=a;1-y=b;1-z=c$, rewrite those equations

$\sqrt{(1-a)(1+b)}+\sqrt{(1+a)(1-b)}=1$;

$\sqrt{(1-b)(1+c)}+\sqrt{(1+b)(1-c)}=\sqrt{2}$

$\sqrt{(1-a)(1+c)}+\sqrt{(1-c)(1+a)}=\sqrt{3}$

and solve for $m/n = (abc)^2 = a^2b^2c^2$

Square both sides and simplify, to get three equations:

$2ab-1=2\sqrt{(1-a^2)(1-b^2)}$

$2bc~ ~ ~ ~ ~ ~=2\sqrt{(1-b^2)(1-c^2)}$

$2ac+1=2\sqrt{(1-c^2)(1-a^2)}$

Square both sides again, and simplify to get three equations:

$a^2+b^2-ab=\frac{3}{4}$

$b^2+c^2~ ~ ~ ~ ~ ~=1$

$a^2+c^2+ac=\frac{3}{4}$

Subtract first and third equation, getting $(b+c)(b-c)=a(b+c)$, $a=b-c$

Put it in first equation, getting $b^2-2bc+c^2+b^2-b(b-c)=b^2+c^2-bc=\frac{3}{4}$, $bc=\frac{1}{4}$

Since $a^2=b^2+c^2-2bc=\frac{1}{2}$, $m/n = a^2b^2c^2 = a^2(bc)^2 = \frac{1}{2}\left(\frac{1}{4}\right)^2=\frac{1}{32}$ and so the final answer is $\boxed{033}$

~bluesoul

Solution 4

Denote $u = 1 - x$, $v = 1 - y$, $w = 1 - z$. Hence, the system of equations given in the problem can be written as \begin{align*} \sqrt{(1-u)(1+v)} + \sqrt{(1+u)(1-v)} & = 1 \hspace{1cm} (1) \\ \sqrt{(1-v)(1+w)} + \sqrt{(1+v)(1-w)} & = \sqrt{2} \hspace{1cm} (2) \\ \sqrt{(1-w)(1+u)} + \sqrt{(1+w)(1-u)} & = \sqrt{3} . \hspace{1cm} (3) \end{align*}

Each equation above takes the following form: \[ \sqrt{(1-a)(1+b)} + \sqrt{(1+a)(1-b)} = k . \]

Now, we simplify this equation by removing radicals.

Denote $p = \sqrt{(1-a)(1+b)}$ and $q = \sqrt{(1+a)(1-b)}$.

Hence, the equation above implies \[ \left\{ \begin{array}{l} p + q = k \\ p^2 = (1-a)(1+b) \\ q^2 = (1+a)(1-b) \end{array} \right.. \]

Hence, $q^2 - p^2 = (1+a)(1-b) - (1-a)(1+b) = 2 (a-b)$. Hence, $q - p = \frac{q^2 - p^2}{p+q} = \frac{2}{k} (a-b)$.

Because $p + q = k$ and $q - p = \frac{2}{k} (a-b)$, we get $q = \frac{a-b}{k} + \frac{k}{2}$. Plugging this into the equation $q^2 = (1+a)(1-b)$ and simplifying it, we get \[ a^2 + \left( k^2 - 2 \right) ab + b^2 = k^2 - \frac{k^4}{4} . \]

Therefore, the system of equations above can be simplified as \begin{align*} u^2 - uv + v^2 & = \frac{3}{4} \\ v^2 + w^2 & = 1 \\ w^2 + wu + u^2 & = \frac{3}{4} . \end{align*}

Denote $w' = - w$. The system of equations above can be equivalently written as \begin{align*} u^2 - uv + v^2 & = \frac{3}{4} \hspace{1cm} (1') \\ v^2 + w'^2 & = 1 \hspace{1cm} (2') \\ w'^2 - w'u + u^2 & = \frac{3}{4} \hspace{1cm} (3') . \end{align*}

Taking $(1') - (3')$, we get \[ (v - w') (v + w' - u) = 0 . \]

Thus, we have either $v - w' = 0$ or $v + w' - u = 0$.

$\textbf{Case 1}$: $v - w' = 0$.

Equation (2') implies $v = w' = \pm \frac{1}{\sqrt{2}}$.

Plugging $v$ and $w'$ into Equation (2), we get contradiction. Therefore, this case is infeasible.

$\textbf{Case 2}$: $v + w' - u = 0$.

Plugging this condition into (1') to substitute $u$, we get \[ v^2 + v w' + w'^2 = \frac{3}{4} \hspace{1cm} (4) . \]

Taking $(4) - (2')$, we get \[ v w' = - \frac{1}{4} . \hspace{1cm} (5) . \]

Taking (4) + (5), we get \[ \left( v + w' \right)^2 = \frac{1}{2} . \]

Hence, $u^2 = \left( v + w' \right)^2 = \frac{1}{2}$.

Therefore, \begin{align*} \left[ (1-x)(1-y)(1-z) \right]^2 & = u^2 (vw)^2 \\ & = u^2 (vw')^2 \\ & = \frac{1}{2} \left( - \frac{1}{4} \right)^2 \\ & = \frac{1}{32} . \end{align*}

Therefore, the answer is $1 + 32 = \boxed{\textbf{(033) }}$.

~Steven Chen (www.professorchenedu.com)

bu-bye \begin{align*} \sin(\alpha + \beta) &= 1/2 \\ \sin(\alpha + \gamma) &= \sqrt2/2 \\ \sin(\beta + \gamma) &= \sqrt3/2. \end{align*} Thus, \begin{align*} \alpha + \beta &= 30^{\circ} \\ \alpha + \gamma &= 45^{\circ} \\ \beta + \gamma &= 60^{\circ}, \end{align*} so $(\alpha, \beta, \gamma) = (15/2^{\circ}, 45/2^{\circ}, 75/2^{\circ})$. Hence,

\[abc = (1-2\sin^2(\alpha))(1-2\sin^2(\beta))(1-2\sin^2(\gamma))=\cos(15^{\circ})\cos(45^{\circ})\cos(75^{\circ})=\frac{\sqrt{2}}{8},\] so $(abc)^2=(\sqrt{2}/8)^2=\frac{1}{32}$, for a final answer of $\boxed{033}$.

Remark

The motivation for the trig substitution is that if $\sin^2(\alpha)=(1-a)/2$, then $\cos^2(\alpha)=(1+a)/2$, 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)

2022 AIME I 15.png

In given equations, $0 \leq x,y,z \leq 2,$ so we define some points: \[\bar {O} = (0, 0), \bar {A} = (1, 0), \bar{M} = \left(\frac {1}{\sqrt{2}},\frac {1}{\sqrt{2}}\right),\] \[\bar {X} = \left(\sqrt {\frac {x}{2}}, \sqrt{1 – \frac{x}{2}}\right), \bar {Y'} = \left(\sqrt {\frac {y}{2}}, \sqrt{1 – \frac{y}{2}}\right),\] \[\bar {Y} = \left(\sqrt {1 – \frac{y}{2}},\sqrt{\frac {y}{2}}\right), \bar {Z} = \left(\sqrt {1 – \frac{z}{2}},\sqrt{\frac {z}{2}}\right).\] Notice, that \[\mid \vec {AO} \mid = \mid \vec {MO} \mid = \mid \vec {XO} \mid =\mid \vec {YO} \mid = \mid \vec {Y'O} \mid =\mid \vec {ZO} \mid = 1\] and each points lies in the first quadrant.

We use given equations and get some scalar products: \[(\vec {XO} \cdot \vec {YO}) = \frac {1}{2} = \cos \angle XOY \implies \angle XOY = 60 ^\circ,\] \[(\vec {XO} \cdot \vec {ZO}) = \frac {\sqrt{3}}{2} = \cos \angle XOZ \implies \angle XOZ = 30^\circ,\] \[(\vec {Y'O} \cdot \vec {ZO}) = \frac {1}{\sqrt{2}} = \cos \angle Y'OZ \implies \angle Y'OZ = 45^\circ.\] So $\angle YOZ = \angle XOY – \angle XOZ = 60 ^\circ – 30 ^\circ = 30 ^\circ, \angle Y'OY = \angle Y'OZ + \angle YOZ = 45^\circ + 30 ^\circ = 75^\circ.$

Points $Y$ and $Y'$ are symmetric with respect to $OM.$

Case 1 \[\angle YOA = \frac{90^\circ – 75^\circ}{2} = 7.5^\circ, \angle ZOA = 30^\circ + 7.5^\circ = 37.5^\circ, \angle XOA = 60^\circ + 7.5^\circ = 67.5^\circ .\] \[1 – x = \left(\sqrt{1 – \frac{x}{2}} \right)^2– \left(\sqrt{\frac {x}{2}}\right)^2 = \sin^2 \angle XOA – \cos^2 \angle XOA = –\cos 2 \angle XOA = –\cos 135^\circ,\] \[1 – y = \cos 15^\circ, 1 – z = \cos 75^\circ \implies \left[ (1–x)(1–y)(1–z) \right]^2 = \left[ \sin 45^\circ \cdot \cos 15^\circ \cdot \sin 15^\circ \right]^2 =\] \[=\left[ \frac {\sin 45^\circ \cdot \sin 30^\circ}{2} \right]^2 = \frac {1}{32} \implies \boxed{\textbf{033}}.\] Case 2

\[\angle Y_1 OA = \frac{90^\circ + 75^\circ}{2} = 82.5^\circ, \angle Z_1 OA = 82.5^\circ – 30^\circ = 52.5^\circ, \angle X_1 OA = 82.5^\circ – 60^\circ = 22.5^\circ \implies \boxed{\textbf{033}}.\]

vladimir.shelomovskii@gmail.com, vvsss

Video Solution

https://youtu.be/i6kDMbav2sk

~Math Gold Medalist


Video Solution

https://youtu.be/aa_VY4e4OOM?si=1lHSwY3v7RICoEpk

~MathProblemSolvingSkills.com


Video Solution

https://www.youtube.com/watch?v=ihKUZ5itcdA

~Steven Chen (www.professorchenedu.com)

See Also

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