Difference between revisions of "2016 IMO Problems/Problem 1"

Latest revision as of 01:00, 15 November 2024

Problem

Triangle $BCF$ has a right angle at $B$ . Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$ . Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$ . Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$ . Let $M$ be the midpoint of $CF$ . Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.

Solution

The Problem shows that $∠ D A C = ∠ D C A = ∠ C A D$ , it follows that $A B ∥ C D$ . Extend $D C$ to intersect $A B$ at $G$ , we get $∠ G F A = ∠ G F B = ∠ C F D$ . Making triangles $△ C D F$ and $△ A G F$ similar. Also, $∠ F D C = ∠ F G A = 90^{\circ}$ and $∠ F B C = 90^{\circ}$ , which points $D$ , $C$ , $B$ , and $F$ are concyclic.

And $∠ B F C = ∠ F B A + ∠ F A B = ∠ F A E = ∠ A F E$ . Triangle $△ A F E$ is congruent to $△ F B M$ , and $A E = E F = F M = M B$ . Let $M X = E A = M F$ , then points $B$ , $C$ , $D$ , $F$ , and $X$ are concyclic.

Finally $A D = D B$ and $∠ D A F = ∠ D B F = ∠ F X D$ . $∠ M F X = ∠ F X D = ∠ F X M$ and $F E ∥ M D$ with $E F = F M = M D = D E$ , making $E F M D$ a rhombus. And $∠ F B D = ∠ M B D = ∠ M X F = ∠ D X F$ and triangle $△ B E M$ is congruent to $△ X E M$ , while $△ M F X$ is congruent to $△ M B D$ which is congruent to $△ F E M$ , so $E M = F X = B D$ .

~Athmyx

Solution 2

Let $\angle FBA = \angle FAB = \angle FAD = \angle FCD = \angle DAE = \angle ADE = \alpha$ . And WLOG, $MF = 1$ . Hence, $CF = 2$ ,

$\implies$ $BF = CF.cos(2\alpha) = 2.cos(2\alpha) = FA$ ,

$\implies$ $DA = \frac{AC}{2cos(\alpha)} = \frac{CF+FA}{2cos(\alpha)} = \frac{2+2cos(2\alpha)}{2cos(\alpha)} = \frac{1+cos(2\alpha)}{cos(\alpha)}$ and

$\implies$ $DE = AE = \frac{DA}{2cos(\alpha)} = \frac{1+cos(2\alpha)}{2.(cos(\alpha))^2} = 1$ .

So $MX = DE = 1$ which means $B$ , $C$ , $X$ and $F$ are concyclic. We know that $DE || MC$ and $DE = 1 = MC$ , so we conclude $MCDE$ is parallelogram. So $\angle AME = \alpha$ . That means $MDEA$ is isosceles trapezoid. Hence, $MD = EA = 1$ . By basic angle chasing,

$\angle MBF = \angle MFB = 2\alpha$ and $\angle MXD = \angle MDX = 2\alpha$ and we have seen that $MB = MF = MD = MX$ , so $BFDX$ is isosceles trapezoid. And we know that $ME$ bisects $\angle FMD$ , so $ME$ is the symmetrical axis of $BFDX$ .

$B$ and $X$ , $D$ and $E$ are symmetrical respect to $ME$ . Hence, the symmetry of $BD$ with respect to $ME$ is $FX$ . And we are done $\blacksquare$ .

~EgeSaribas

@@ Line 15: / Line 15: @@
 Finally  $A D = D B$  and  $∠ D A F = ∠ D B F = ∠ F X D$ .   $∠ M F X = ∠ F X D = ∠ F X M$  and  $F E ∥ M D$  with  $E F = F M = M D = D E$ , making  $E F M D$  a rhombus. And  $∠ F B D = ∠ M B D = ∠ M X F = ∠ D X F$  and triangle  $△ B E M$  is congruent to  $△ X E M$ , while  $△ M F X$  is congruent to  $△ M B D$  which is congruent to  $△ F E M$ , so  $E M = F X = B D$ .
-~Athmyx
+~[https://artofproblemsolving.com/wiki/index.php/User:Athmyx Athmyx]
 == Solution 2 ==
-Let <math>\angle FBA = \angle FAB = \angle FAD = \angle FCD = \alpha</math>. And WLOG, <math>MF = 1</math>. Hence, <math>CF = 2</math>,
+Let <math>\angle FBA = \angle FAB = \angle FAD = \angle FCD = \angle DAE = \angle ADE = \alpha</math>. And WLOG, <math>MF = 1</math>. Hence, <math>CF = 2</math>,
-<math>\implies</math> <math>BF = 2.cos(2\alpha) = FA</math>,
+<math>\implies</math> <math>BF = CF.cos(2\alpha) = 2.cos(2\alpha) = FA</math>,
-<math>\implies</math> <math>DA = \frac{AC}{2cos(\alpha)} = \frac{1+cos(2\alpha)}{cos(\alpha)}</math> and
+<math>\implies</math> <math>DA = \frac{AC}{2cos(\alpha)} = \frac{CF+FA}{2cos(\alpha)} = \frac{2+2cos(2\alpha)}{2cos(\alpha)} = \frac{1+cos(2\alpha)}{cos(\alpha)}</math> and
 <math>\implies</math> <math>DE = AE = \frac{DA}{2cos(\alpha)} = \frac{1+cos(2\alpha)}{2.(cos(\alpha))^2} = 1</math>.
-So <math>MX = DE = 1</math> which means <math>B</math>, <math>C</math>, <math>X</math> and <math>F</math> are concyclic. We know that <math>DE || MC</math> and <math>DE = 1 = MC</math>, so we conclude <math>MCDE</math> is parallelogram. So <math>\angle AME = \alpha</math>. That means <math>MDEA</math> is isosceles trapezoid. Hence, <math>MD = EA = 1</math>. By basic angle chasing,
+So <math>MX = DE = 1</math> which means <math>B</math>, <math>C</math>, <math>X</math> and <math>F</math> are concyclic. We know that <math>DE  ||  MC</math> and <math>DE = 1 = MC</math>, so we conclude <math>MCDE</math> is parallelogram. So <math>\angle AME = \alpha</math>. That means <math>MDEA</math> is isosceles trapezoid. Hence, <math>MD = EA = 1</math>. By basic angle chasing,
 <math>\angle MBF = \angle MFB = 2\alpha</math> and <math>\angle MXD = \angle MDX = 2\alpha</math> and we have seen that <math>MB = MF = MD = MX</math>, so <math>BFDX</math> is isosceles trapezoid. And we know that <math>ME</math> bisects <math>\angle FMD</math>, so <math>ME</math> is the symmetrical axis of <math>BFDX</math>.
-İt is clear that the symmetry of <math>BD</math> with respect to <math>ME</math> is <math>FX</math>. And we are done <math>\blacksquare</math>.
+<math>B</math> and <math>X</math>, <math>D</math> and <math>E</math> are symmetrical respect to <math>ME</math>. Hence, the symmetry of <math>BD</math> with respect to <math>ME</math> is <math>FX</math>. And we are done <math>\blacksquare</math>.
 ~EgeSaribas

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Difference between revisions of "2016 IMO Problems/Problem 1"

Latest revision as of 01:00, 15 November 2024

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Problem

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Solution 2

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