Difference between revisions of "1982 IMO Problems/Problem 5"

Revision as of 05:20, 15 June 2024

Problem

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that ${AM\over AC}={CN\over CE}=r.$ Determine $r$ if $B,M$ and $N$ are collinear.

Solution 1

O is the center of the regular hexagon. Then we clearly have $ABC\cong COA\cong EOC$ . And therefore we have also obviously $ABM\cong AOM\cong CON$ , as $\frac{AM}{AC} =\frac{CN}{CE}$ . So we have $\angle{BMA} =\angle{AMO} =\angle{CNO}$ and $\angle{NOC} =\angle{ABM}$ . Because of $\angle{AMO} =\angle{CNO}$ the quadrilateral $ONCM$ is cyclic. $\Rightarrow \angle{NOC} =\angle{NMC} =\angle{BMA}$ . And as we also have $\angle{NOC} =\angle{ABM}$ we get $\angle{ABM} =\angle{BMA}$ . $\Rightarrow AB=AM$ . And as $AC=\sqrt{3} \cdot AB$ we get $r=\frac{AM}{AC} =\frac{AB}{\sqrt{3} \cdot AB} =\frac{1}{\sqrt{3}}$ .

This solution was posted and copyrighted by Kathi. The original thread for this problem can be found here: [1]

Solution 2

Let $X$ be the intersection of $AC$ and $BE$ . $X$ is the mid-point of $AC$ . Since $B$ , $M$ , and $N$ are collinear, then by Menelaus Theorem, $\frac{CN}{NE}\cdot\frac{EB}{BX}\cdot\frac{XM}{MC}=1$ . Let the sidelength of the hexagon be $1$ . Then $AC=CE=\sqrt{3}$ . $\frac{CN}{NE}=\frac{CN}{CE-CN}=\frac{\frac{CN}{CE}}{1-\frac{CN}{CE}}=\frac{r}{1-r}$ $\frac{EB}{BX}=\frac{2}{\frac{1}{2}}=4$ $\frac{XM}{MC}=\frac{AM-AX}{AC-AM}=\frac{\frac{AM}{AC}-\frac{AX}{AC}}{1-\frac{AM}{AC}}=\frac{r-\frac{1}{2}}{1-r}$ Substituting them into the first equation yields $\frac{r}{1-r}\cdot\frac{4}{1}\cdot\frac{r-\frac{1}{2}}{1-r}=1$ $3r^2=1$ $\therefore r=\frac{\sqrt{3}}{3}$

This solution was posted and copyrighted by leepakhin. The original thread for this problem can be found here: [2]

Solution 3

Note $x=m(\widehat {EBM})$ . From the relation $r=\frac{AM}{AC}=\frac{CN}{CE}$ results $\frac{MA}{MC}=\frac{NC}{NE}$ , i.e.

$\frac{BA}{BC}\cdot \frac{\sin (60+x)}{\sin (60-x)}=\frac{BC}{BE}\cdot \frac{\sin (60-x)}{\sin x}$ . Thus, $2\sin x\sin (60+x)=\sin^2(60-x)\Longrightarrow$ $2[\cos 60-\cos (60+2x)]=1-\cos (120-2x)\Longrightarrow \cos (60+2x)=0\Longrightarrow x=15^{\circ}.$

Therefore, $\frac{MA}{MC}=\frac{\sin 75}{\sin 45}=\frac{1+\sqrt 3}{2}$ , i.e. $r=\frac{MA}{AC}=\frac{1+\sqrt 3}{3+\sqrt 3}=\frac{1}{\sqrt 3}\Longrightarrow r=\frac{\sqrt 3}{3}<\math>

This solution was posted and copyrighted by Virgil. The original thread for this problem can be found hercommunity/p398343]

Let <math>AM = CN = a$ (Error compiling LaTeX. Unknown error_msg). By the cosine rule,

$AC = \sqrt{AB^{2} + BC^{2} - 2 \cdot AB \cdot BC \cdot cos \angle BAC}

= \sqrt{1 + 1 - 2 cos 120^{\circ}}

= \sqrt{3}$ (Error compiling LaTeX. Unknown error_msg).

$BM = \sqrt{a^{2} + 1 - 2a \cdot cos 30^{\circ}}

= \sqrt{a^{2} - \sqrt{3} \cdot a + 1}$ (Error compiling LaTeX. Unknown error_msg)

$MN = \sqrt{(\sqrt{3} - a)^{2} + a^{2} - 2 \cdot (\sqrt{3} - a) \cdot a \cdot cos \angle MCN}

= \sqrt{3 + a^{2} - 2\cdot \sqrt{3} \cdot a + a^{2} - \sqrt{3} \cdot a + a^{2}}

= \sqrt{3a^{2} - 3\sqrt{3}\cdot a + 3}

= BM \cdot \sqrt{3}$ (Error compiling LaTeX. Unknown error_msg).

Now if B, M, and N are collinear, then $\angle AMB = \angle CMN

\implies sin \angle AMB = sin \angle CMN$ (Error compiling LaTeX. Unknown error_msg).

By the law of Sines,

$\frac{1}{sin \angle AMB} = \frac{BM}{sin 30^{\circ}} = 2BM

\implies sin \angle AMB = \frac{1}{2BM}$ (Error compiling LaTeX. Unknown error_msg).

Also,

$\frac{a}{sin \angle CMN} = \frac{\sqrt{3} \cdot BM}{sin 60^{\circ}} = 2BM

\implies sin \angle CMN = \frac{a}{2BM}$ (Error compiling LaTeX. Unknown error_msg).

But $sin \angle AMB = sin \angle CMN

\implies \frac{1}{2BM} = \frac{a}{2BM}$ (Error compiling LaTeX. Unknown error_msg), which means $a = 1$ . So, r = \frac{1}{\sqrt{3}} $.

@@ Line 29: / Line 29: @@
 <math>2[\cos 60-\cos (60+2x)]=1-\cos (120-2x)\Longrightarrow \cos (60+2x)=0\Longrightarrow x=15^{\circ}.</math>
-Therefore, <math>\frac{MA}{MC}=\frac{\sin 75}{\sin 45}=\frac{1+\sqrt 3}{2}</math>, i.e. <math>r=\frac{MA}{AC}=\frac{1+\sqrt 3}{3+\sqrt 3}=\frac{1}{\sqrt 3}\Longrightarrow r=\frac{\sqrt 3}{3}.</math>
+Therefore, <math>\frac{MA}{MC}=\frac{\sin 75}{\sin 45}=\frac{1+\sqrt 3}{2}</math>, i.e. <math>r=\frac{MA}{AC}=\frac{1+\sqrt 3}{3+\sqrt 3}=\frac{1}{\sqrt 3}\Longrightarrow r=\frac{\sqrt 3}{3}<\math>
-This solution was posted and copyrighted by Virgil Nicula. The original thread for this problem can be found here: [https://aops.com/community/p398343]
+This solution was posted and copyrighted by Virgil. The original thread for this problem can be found hercommunity/p398343]
-==Solution 4==
 Let <math>AM = CN = a </math>. By the cosine rule,
 <math>AC = \sqrt{AB^{2} + BC^{2} - 2 \cdot AB \cdot BC \cdot cos \angle BAC}
 = \sqrt{1 + 1 - 2 cos 120^{\circ}}
 = \sqrt{3} </math>.
 <math>BM = \sqrt{a^{2} + 1 - 2a \cdot cos 30^{\circ}}
 = \sqrt{a^{2} - \sqrt{3} \cdot a + 1} </math>
 <math>MN = \sqrt{(\sqrt{3} - a)^{2} + a^{2} - 2 \cdot (\sqrt{3} - a) \cdot a \cdot cos \angle MCN}
 = \sqrt{3 + a^{2} - 2\cdot \sqrt{3} \cdot a + a^{2} - \sqrt{3} \cdot a + a^{2}}
 = \sqrt{3a^{2} - 3\sqrt{3}\cdot a + 3}
 = BM \cdot \sqrt{3} </math>.
 Now if B, M, and N are collinear, then <math>\angle AMB = \angle CMN
 \implies sin \angle AMB = sin \angle CMN </math>.
 By the law of Sines,
 <math>\frac{1}{sin \angle AMB} = \frac{BM}{sin 30^{\circ}} = 2BM
 \implies sin \angle AMB = \frac{1}{2BM} </math>.
 Also,
 <math>\frac{a}{sin \angle CMN} = \frac{\sqrt{3} \cdot BM}{sin 60^{\circ}} = 2BM
 \implies sin \angle CMN = \frac{a}{2BM} </math>.
 But <math>sin \angle AMB = sin \angle CMN
 \implies \frac{1}{2BM} = \frac{a}{2BM} </math>, which means <math>a = 1 </math>. So, r = \frac{1}{\sqrt{3}} $.
-== See Also == {{IMO box|year=1982|num-b=4|num-a=6}}

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