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Difference between revisions of "1991 IMO Problems/Problem 5"

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Let <math> \,ABC\,</math> be a triangle and <math> \,P\,</math> an interior point of <math> \,ABC\,</math>. Show that at least one of the angles <math> \,\angle PAB,\;\angle PBC,\;\angle PCA\,</math> is less than or equal to <math> 30^{\circ }</math>.
 
Let <math> \,ABC\,</math> be a triangle and <math> \,P\,</math> an interior point of <math> \,ABC\,</math>. Show that at least one of the angles <math> \,\angle PAB,\;\angle PBC,\;\angle PCA\,</math> is less than or equal to <math> 30^{\circ }</math>.
  
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== Solution ==
+
== Solution 1 ==
−
Let <math>A_{1}</math> , <math>A_{2}</math>, and <math>A_{3}</math> be <math>\measuredangle CAB</math>, <math>\measuredangle ABC</math>, <math>\measuredangle BCA</math>, respcetively.
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Let <math>A_{1}</math> , <math>A_{2}</math>, and <math>A_{3}</math> be <math>\angle CAB</math>, <math>\angle ABC</math>, <math>\angle BCA</math>, respectively.
  
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Let <math>\alpha_{1}</math> , <math>\alpha_{2}</math>, and <math>\alpha_{3}</math> be <math>\measuredangle PAB</math>, <math>\measuredangle PBC</math>, <math>\measuredangle PCA</math>, respcetively.
+
Let <math>\alpha_{1}</math> , <math>\alpha_{2}</math>, and <math>\alpha_{3}</math> be <math>\angle PAB</math>, <math>\angle PBC</math>, <math>\angle PCA</math>, respcetively.
  
 
Using law of sines on <math>\Delta PAB</math> we get: <math>\frac{\left| PA \right|}{sin(A_{2}-\alpha_{2})}=\frac{\left| PB \right|}{sin(\alpha_{1})}</math>, therefore, <math>\frac{\left| PA \right|}{\left| PB \right|}=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}</math>
 
Using law of sines on <math>\Delta PAB</math> we get: <math>\frac{\left| PA \right|}{sin(A_{2}-\alpha_{2})}=\frac{\left| PB \right|}{sin(\alpha_{1})}</math>, therefore, <math>\frac{\left| PA \right|}{\left| PB \right|}=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}</math>
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Using law of sines on <math>\Delta PCA</math> we get: <math>\frac{\left| PC \right|}{sin(A_{1}-\alpha_{1})}=\frac{\left| PA \right|}{sin(\alpha_{3})}</math>, therefore, <math>\frac{\left| PC \right|}{\left| PA \right|}=\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}</math>
 
Using law of sines on <math>\Delta PCA</math> we get: <math>\frac{\left| PC \right|}{sin(A_{1}-\alpha_{1})}=\frac{\left| PA \right|}{sin(\alpha_{3})}</math>, therefore, <math>\frac{\left| PC \right|}{\left| PA \right|}=\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}</math>
 +
 +
Multiply all three equations we get:
 +
<math>\frac{\left| PA \right|}{\left| PB \right|}\frac{\left| PB \right|}{\left| PC \right|}\frac{\left| PC \right|}{\left| PA \right|}=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}\frac{sin(A_{3}-\alpha_{3})}{sin(\alpha_{2})}\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}</math>
 +
 +
<math>1=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}\frac{sin(A_{3}-\alpha_{3})}{sin(\alpha_{2})}\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}</math>
 +
 +
<math>\prod_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}=1</math>
 +
 +
Using AM-GM we get:
 +
 +
<math>\frac{1}{3}\sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}\ge \sqrt[3]{\prod_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}}</math>
 +
 +
<math>\frac{1}{3}\sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}\ge 1</math>
 +
 +
<math>\sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}\ge 3</math>.  [Inequality 1]
 +
 +
<math>\sum_{i=1}^{3}\frac{sin(A_{i})cos(\alpha_{i})-cos(A_{i})sin(\alpha_{i})}{sin(\alpha_{i})}\ge 3</math>
 +
 +
<math>\sum_{i=1}^{3}\left[ sin(A_{i})cot(\alpha_{i})-cos(A_{i})\right]\ge 3</math>
 +
 +
Note that for <math>0<\alpha_{i}<180^{\circ}</math>, <math>cot(\alpha_{i})</math> decreases with increasing <math>\alpha_{i}</math> and fixed <math>A_{i}</math>
 +
 +
Therefore, <math>\left[ sin(A_{i})cot(\alpha_{i})-cos(A_{i})\right]</math> decreases with increasing <math>\alpha_{i}</math> and fixed <math>A_{i}</math>
 +
 +
From trigonometric identity:
 +
 +
<math>sin(x)+sin(y)=2sin\left( \frac{x+y}{2} \right)cos\left( \frac{x-y}{2} \right)</math>,
 +
 +
since <math>-1\le cos\left( \frac{x-y}{2} \right) \le 1</math>, then:
 +
 +
<math>sin(x)+sin(y) \le 2sin\left( \frac{x+y}{2} \right)</math>
 +
 +
Therefore,
 +
 +
<math>sin(A_{1}-30^{\circ})+sin(A_{2}-30^{\circ}) \le 2sin\left( \frac{A_{1}+A_{2}-60^{\circ}}{2} \right)</math>
 +
 +
and also,
 +
 +
<math>sin(A_{3}-30^{\circ})+sin(30^{\circ}) \le 2sin\left( \frac{A_{3}}{2} \right)</math>
 +
 +
Adding these two inequalities we get:
 +
 +
<math>sin(30^{\circ})+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 2\left[ sin\left( \frac{A_{1}+A_{2}-60^{\circ}}{2} \right)+sin\left( \frac{A_{3}}{2} \right) \right]</math>
 +
 +
<math>\frac{1}{2}+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 2\left[ 2sin\left( \frac{A_{1}+A_{2}+A_{3}-60^{\circ}}{4} \right) \right]</math>
 +
 +
<math>\frac{1}{2}+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 2\left[ 2sin\left( \frac{180^{\circ}-60^{\circ}}{4} \right) \right]</math>
 +
 +
<math>\frac{1}{2}+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 4sin\left( 30^{\circ} \right)</math>
 +
 +
<math>\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le \frac{3}{2}</math>
 +
 +
<math>2\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 3</math>.
 +
 +
<math>\sum_{i=1}^{3}\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le 3</math>. [Inequality 2]
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 +
Combining [Inequality 1] and [Inequality 2] we see the following:
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 +
<math>\sum_{i=1}^{3}\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}</math>
 +
 +
This implies that for at least one of the values of <math>i=1</math>,<math>2</math>,or <math>3</math>, the following is true:
 +
 +
<math>\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}</math>
 +
 +
or
 +
 +
<math>\frac{sin(\alpha_{i})}{sin(A_{i}-\alpha_{i})}\le \frac{sin(30^{\circ})}{sin(A_{i}-30^{\circ})}</math>
 +
 +
Which means that for at least one of the values of <math>i=1</math>,<math>2</math>,or <math>3</math>, the following is true:
 +
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<math>\alpha_{i} \le 30^{\circ}</math>
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 +
Therefore, at least one of the angles <math> \,\angle PAB,\;\angle PBC,\;\angle PCA\,</math> is less than or equal to <math> 30^{\circ }</math>.
 +
 +
~Tomas Diaz, orders@tomasdiaz.com
  
 
{{alternate solutions}}
 
{{alternate solutions}}
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 +
== Solution 2 ==
 +
 +
At least one of <math>\angle ABC, \angle BCA, \angle CAB \ge 60^\circ</math>. Without loss of generality, assume that <math>\angle BCA \ge 60^\circ</math>
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If <math>\angle PAB > 30^\circ</math> and <math>\angle PBC > 30^\circ</math>
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 +
Draw a circle <math>R</math> centered at <math>O</math> and passing through <math>A, P, B</math>. Since <math>P</math> is an interior point of <math>\triangle ABC</math>, thus <math>C</math> is outside the circle <math>R</math>
 +
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Draw two lines <math>CD, CE</math> passing through <math>C</math> and tangent to <math>R</math>. Line <math>CD</math> intersect <math>R</math> at <math>D</math>, and line <math>CE</math> intersect <math>R</math> at <math>E</math>. Choose <math>D</math> near <math>A</math>, and choose <math>E</math> near <math>B</math>
 +
 +
Extends line <math>BC</math>, and intersect <math>R</math> at <math>F</math> other than <math>B</math> when <math>BC</math> is not tangent to <math>R</math>. If <math>BC</math> is tangent to <math>R</math>, we have <math>B = E</math> be the tangent point, and simply let <math>F = B = E</math>
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Draw the segment <math>OE</math>, and choose a point <math>G</math> on <math>R</math> such that <math>\angle GOE = 60^\circ</math>. There are two possible points, we choose <math>G</math> near point <math>P</math>. Draw segments <math>OG, GE</math>, thus <math>\triangle GOE</math> is an equilateral triangle
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 +
Draw segments <math>OP, OC, OB, OF, PB, GC</math>
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<math>\angle OCE = \dfrac{1}{2} \angle DCE \ge \dfrac{1}{2} \angle BCA \ge 30^\circ</math>. Then we have <math>\angle COE = 90^\circ - \angle OCE \le 60^\circ = \angle GOE</math>
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<math>\angle POB = 2 \angle PAB > 60^\circ, \angle POF = 2 \angle PBC > 60^\circ</math>, since we have either <math>\angle POE \ge \angle POB</math> or <math>\angle POE \ge \angle POF</math>, thus <math>\angle POE > 60^\circ = \angle GOE</math>
 +
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Thus we have <math>\angle COE \le \angle GOE < \angle POE</math>, then <math>\angle OCE \le \angle GCE < \angle PCE</math>
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Because <math>\angle GCE \ge \angle OCE \ge 30^\circ = \angle GEC</math>, thus <math>GC \le GE = OG</math>, and <math>\angle GCO \ge \angle GOC</math>
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Finally, <math>\angle PCA = \angle ACE - \angle PCE < \angle ACE - \angle GCE = \angle ACO - \angle GCO</math>
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Since <math>\angle ACO \le \angle DCO</math>, and <math>\angle GCO \ge \angle GOC</math>, thus we have <math>\angle PCA < \angle ACO - \angle GCO \le \angle DCO - \angle GOC = 90^\circ - \angle COE - \angle GOC = 90^\circ - \angle GOE = 30^\circ</math>
 +
 +
We have proved that when <math>\angle PAB > 30^\circ</math> and <math>\angle PBC > 30^\circ</math>, the angle <math>\angle PCA</math> must be less than <math>30^\circ</math>.  Thus at least one of <math>\angle PAB, \angle PBC, \angle PCA</math> should less than or equal to <math>30^\circ</math>
 +
 +
~Joseph Tsai, mgtsai@gmail.com
 +
 +
==See Also==
 +
 +
{{IMO box|year=1991|num-b=4|num-a=6}}
 +
[[Category:Olympiad Geometry Problems]]
 +
[[Category:Geometry Problems]]

Latest revision as of 04:01, 23 January 2024

Problem

Let $\,ABC\,$ be a triangle and $\,P\,$ an interior point of $\,ABC\,$. Show that at least one of the angles $\,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $30^{\circ }$.

Solution 1

Let $A_{1}$ , $A_{2}$, and $A_{3}$ be $\angle CAB$, $\angle ABC$, $\angle BCA$, respectively.

Let $\alpha_{1}$ , $\alpha_{2}$, and $\alpha_{3}$ be $\angle PAB$, $\angle PBC$, $\angle PCA$, respcetively.

Using law of sines on $\Delta PAB$ we get: $\frac{\left| PA \right|}{sin(A_{2}-\alpha_{2})}=\frac{\left| PB \right|}{sin(\alpha_{1})}$, therefore, $\frac{\left| PA \right|}{\left| PB \right|}=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}$

Using law of sines on $\Delta PBC$ we get: $\frac{\left| PB \right|}{sin(A_{3}-\alpha_{3})}=\frac{\left| PC \right|}{sin(\alpha_{2})}$, therefore, $\frac{\left| PB \right|}{\left| PC \right|}=\frac{sin(A_{3}-\alpha_{3})}{sin(\alpha_{2})}$

Using law of sines on $\Delta PCA$ we get: $\frac{\left| PC \right|}{sin(A_{1}-\alpha_{1})}=\frac{\left| PA \right|}{sin(\alpha_{3})}$, therefore, $\frac{\left| PC \right|}{\left| PA \right|}=\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}$

Multiply all three equations we get: $\frac{\left| PA \right|}{\left| PB \right|}\frac{\left| PB \right|}{\left| PC \right|}\frac{\left| PC \right|}{\left| PA \right|}=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}\frac{sin(A_{3}-\alpha_{3})}{sin(\alpha_{2})}\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}$

$1=\frac{sin(A_{2}-\alpha_{2})}{sin(\alpha_{1})}\frac{sin(A_{3}-\alpha_{3})}{sin(\alpha_{2})}\frac{sin(A_{1}-\alpha_{1})}{sin(\alpha_{3})}$

$\prod_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}=1$

Using AM-GM we get:

$\frac{1}{3}\sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}\ge \sqrt[3]{\prod_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}}$

$\frac{1}{3}\sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}\ge 1$

$\sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}\ge 3$. [Inequality 1]

$\sum_{i=1}^{3}\frac{sin(A_{i})cos(\alpha_{i})-cos(A_{i})sin(\alpha_{i})}{sin(\alpha_{i})}\ge 3$

$\sum_{i=1}^{3}\left[ sin(A_{i})cot(\alpha_{i})-cos(A_{i})\right]\ge 3$

Note that for $0<\alpha_{i}<180^{\circ}$, $cot(\alpha_{i})$ decreases with increasing $\alpha_{i}$ and fixed $A_{i}$

Therefore, $\left[ sin(A_{i})cot(\alpha_{i})-cos(A_{i})\right]$ decreases with increasing $\alpha_{i}$ and fixed $A_{i}$

From trigonometric identity:

$sin(x)+sin(y)=2sin\left( \frac{x+y}{2} \right)cos\left( \frac{x-y}{2} \right)$,

since $-1\le cos\left( \frac{x-y}{2} \right) \le 1$, then:

$sin(x)+sin(y) \le 2sin\left( \frac{x+y}{2} \right)$

Therefore,

$sin(A_{1}-30^{\circ})+sin(A_{2}-30^{\circ}) \le 2sin\left( \frac{A_{1}+A_{2}-60^{\circ}}{2} \right)$

and also,

$sin(A_{3}-30^{\circ})+sin(30^{\circ}) \le 2sin\left( \frac{A_{3}}{2} \right)$

Adding these two inequalities we get:

$sin(30^{\circ})+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 2\left[ sin\left( \frac{A_{1}+A_{2}-60^{\circ}}{2} \right)+sin\left( \frac{A_{3}}{2} \right) \right]$

$\frac{1}{2}+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 2\left[ 2sin\left( \frac{A_{1}+A_{2}+A_{3}-60^{\circ}}{4} \right) \right]$

$\frac{1}{2}+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 2\left[ 2sin\left( \frac{180^{\circ}-60^{\circ}}{4} \right) \right]$

$\frac{1}{2}+\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 4sin\left( 30^{\circ} \right)$

$\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le \frac{3}{2}$

$2\sum_{i=1}^{3}sin(A_{i}-30^{\circ})\le 3$.

$\sum_{i=1}^{3}\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le 3$. [Inequality 2]

Combining [Inequality 1] and [Inequality 2] we see the following:

$\sum_{i=1}^{3}\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}$

This implies that for at least one of the values of $i=1$,$2$,or $3$, the following is true:

$\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}$

or

$\frac{sin(\alpha_{i})}{sin(A_{i}-\alpha_{i})}\le \frac{sin(30^{\circ})}{sin(A_{i}-30^{\circ})}$

Which means that for at least one of the values of $i=1$,$2$,or $3$, the following is true:

$\alpha_{i} \le 30^{\circ}$

Therefore, at least one of the angles $\,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $30^{\circ }$.

~Tomas Diaz, orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Solution 2

At least one of $\angle ABC, \angle BCA, \angle CAB \ge 60^\circ$. Without loss of generality, assume that $\angle BCA \ge 60^\circ$

If $\angle PAB > 30^\circ$ and $\angle PBC > 30^\circ$

Draw a circle $R$ centered at $O$ and passing through $A, P, B$. Since $P$ is an interior point of $\triangle ABC$, thus $C$ is outside the circle $R$

Draw two lines $CD, CE$ passing through $C$ and tangent to $R$. Line $CD$ intersect $R$ at $D$, and line $CE$ intersect $R$ at $E$. Choose $D$ near $A$, and choose $E$ near $B$

Extends line $BC$, and intersect $R$ at $F$ other than $B$ when $BC$ is not tangent to $R$. If $BC$ is tangent to $R$, we have $B = E$ be the tangent point, and simply let $F = B = E$

Draw the segment $OE$, and choose a point $G$ on $R$ such that $\angle GOE = 60^\circ$. There are two possible points, we choose $G$ near point $P$. Draw segments $OG, GE$, thus $\triangle GOE$ is an equilateral triangle

Draw segments $OP, OC, OB, OF, PB, GC$

$\angle OCE = \dfrac{1}{2} \angle DCE \ge \dfrac{1}{2} \angle BCA \ge 30^\circ$. Then we have $\angle COE = 90^\circ - \angle OCE \le 60^\circ = \angle GOE$

$\angle POB = 2 \angle PAB > 60^\circ, \angle POF = 2 \angle PBC > 60^\circ$, since we have either $\angle POE \ge \angle POB$ or $\angle POE \ge \angle POF$, thus $\angle POE > 60^\circ = \angle GOE$

Thus we have $\angle COE \le \angle GOE < \angle POE$, then $\angle OCE \le \angle GCE < \angle PCE$

Because $\angle GCE \ge \angle OCE \ge 30^\circ = \angle GEC$, thus $GC \le GE = OG$, and $\angle GCO \ge \angle GOC$

Finally, $\angle PCA = \angle ACE - \angle PCE < \angle ACE - \angle GCE = \angle ACO - \angle GCO$

Since $\angle ACO \le \angle DCO$, and $\angle GCO \ge \angle GOC$, thus we have $\angle PCA < \angle ACO - \angle GCO \le \angle DCO - \angle GOC = 90^\circ - \angle COE - \angle GOC = 90^\circ - \angle GOE = 30^\circ$

We have proved that when $\angle PAB > 30^\circ$ and $\angle PBC > 30^\circ$, the angle $\angle PCA$ must be less than $30^\circ$. Thus at least one of $\angle PAB, \angle PBC, \angle PCA$ should less than or equal to $30^\circ$

~Joseph Tsai, mgtsai@gmail.com

See Also

1991 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
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