Difference between revisions of "1961 IMO Problems/Problem 4"

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In the interior of [[triangle]] <math>P_1P_2P_3</math> a [[point]] <math>P</math> is given.  Let <math>Q_1,Q_2,Q_3</math> be the [[intersection]]s of <math>PP_1, PP_2,PP_3</math> with the opposing [[edge]]s of triangle <math>P_1P_2P_3</math>.  Prove that among the [[ratio]]s <math>\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}</math> there exists one not larger than <math>2</math> and one not smaller than <math>2</math>.
 
In the interior of [[triangle]] <math>P_1P_2P_3</math> a [[point]] <math>P</math> is given.  Let <math>Q_1,Q_2,Q_3</math> be the [[intersection]]s of <math>PP_1, PP_2,PP_3</math> with the opposing [[edge]]s of triangle <math>P_1P_2P_3</math>.  Prove that among the [[ratio]]s <math>\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}</math> there exists one not larger than <math>2</math> and one not smaller than <math>2</math>.
  
==Solution==
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==Solution 1==
{{solution}}
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Let <math>[ABC]</math> denote the area of triangle <math>ABC</math>.
  
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Since triangles <math>P_1P_2P_3</math> and <math>PP_2P_3</math> share the base <math>P_2P_3</math>, we have <math>\frac{[PP_2P_3]}{[P_1P_2P_3]}=\frac{PQ_1}{P_1Q_1}</math>.
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Similarly, <math>\frac{[PP_1P_3]}{[P_1P_2P_3]}=\frac{PQ_2}{P_2Q_2}, \frac{[PP_1P_2]}{[P_1P_2P_3]}=\frac{PQ_3}{P_3Q_3}</math>.
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Adding all of these gives <math>\frac{[PP_1P_3]}{[P_1P_2P_3]}+\frac{[PP_1P_2]}{[P_1P_2P_3]}+\frac{[PP_2P_3]}{[P_1P_2P_3]}=\frac{PQ_2}{P_2Q_2}+\frac{PQ_3}{P_3Q_3}+\frac{PQ_1}{P_1Q_1}=1</math>.
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We see that we must have at least one of the three fractions not greater than <math>\frac{1}{3}</math>, and at least one not less than <math>\frac{1}{3}</math>. These correspond to ratios <math>\frac{PP_i}{PQ_i}</math> being less than or equal to <math>2</math>, and greater than or equal to <math>2</math>, respectively, so we are done.
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==Solution 2==
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Let <math>K_1=[P_2PP_3], K_2=[P_3PP_1],</math> and <math>K_3=[P_1PP_2].</math>  Note that by same base in triangles <math>P_2PP_3</math> and <math>P_2P_1P_3,</math> <cmath>\frac{P_1P}{PQ_1}+1=\frac{P_1Q_1}{PQ_1}=\frac{[P_2P_1P_3]}{[P_2PP_3]}=\frac{K_1+K_2+K_3}{K_1}.</cmath>  Thus,    <cmath>\frac{P_1P}{PQ_1}=\frac{K_2+K_3}{K_1}</cmath><cmath>\frac{P_2P}{PQ_2}=\frac{K_1+K_3}{K_2}</cmath> <cmath>\frac{P_3P}{PQ_3}=\frac{K_1+K_2}{K_3}.</cmath>
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Without loss of generality, assume <math>K_1\leq K_2\leq K_3.</math> Hence, <cmath>\frac{P_1P}{PQ_1}=\frac{K_2+K_3}{K_1}\geq \frac{K_1+K_1}{K_1}=2</cmath> and <cmath>\frac{P_3P}{PQ_3}=\frac{K_1+K_2}{K+3}\leq \frac{K_3+K_3}{K_3}=2,</cmath> as desired. <math>\blacksquare</math>
  
 
{{IMO box|year=1961|num-b=3|num-a=5}}
 
{{IMO box|year=1961|num-b=3|num-a=5}}

Latest revision as of 20:04, 5 December 2018

Problem

In the interior of triangle $P_1P_2P_3$ a point $P$ is given. Let $Q_1,Q_2,Q_3$ be the intersections of $PP_1, PP_2,PP_3$ with the opposing edges of triangle $P_1P_2P_3$. Prove that among the ratios $\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}$ there exists one not larger than $2$ and one not smaller than $2$.

Solution 1

Let $[ABC]$ denote the area of triangle $ABC$.

Since triangles $P_1P_2P_3$ and $PP_2P_3$ share the base $P_2P_3$, we have $\frac{[PP_2P_3]}{[P_1P_2P_3]}=\frac{PQ_1}{P_1Q_1}$.

Similarly, $\frac{[PP_1P_3]}{[P_1P_2P_3]}=\frac{PQ_2}{P_2Q_2}, \frac{[PP_1P_2]}{[P_1P_2P_3]}=\frac{PQ_3}{P_3Q_3}$.

Adding all of these gives $\frac{[PP_1P_3]}{[P_1P_2P_3]}+\frac{[PP_1P_2]}{[P_1P_2P_3]}+\frac{[PP_2P_3]}{[P_1P_2P_3]}=\frac{PQ_2}{P_2Q_2}+\frac{PQ_3}{P_3Q_3}+\frac{PQ_1}{P_1Q_1}=1$.

We see that we must have at least one of the three fractions not greater than $\frac{1}{3}$, and at least one not less than $\frac{1}{3}$. These correspond to ratios $\frac{PP_i}{PQ_i}$ being less than or equal to $2$, and greater than or equal to $2$, respectively, so we are done.

Solution 2

Let $K_1=[P_2PP_3], K_2=[P_3PP_1],$ and $K_3=[P_1PP_2].$ Note that by same base in triangles $P_2PP_3$ and $P_2P_1P_3,$ \[\frac{P_1P}{PQ_1}+1=\frac{P_1Q_1}{PQ_1}=\frac{[P_2P_1P_3]}{[P_2PP_3]}=\frac{K_1+K_2+K_3}{K_1}.\] Thus, \[\frac{P_1P}{PQ_1}=\frac{K_2+K_3}{K_1}\]\[\frac{P_2P}{PQ_2}=\frac{K_1+K_3}{K_2}\] \[\frac{P_3P}{PQ_3}=\frac{K_1+K_2}{K_3}.\] Without loss of generality, assume $K_1\leq K_2\leq K_3.$ Hence, \[\frac{P_1P}{PQ_1}=\frac{K_2+K_3}{K_1}\geq \frac{K_1+K_1}{K_1}=2\] and \[\frac{P_3P}{PQ_3}=\frac{K_1+K_2}{K+3}\leq \frac{K_3+K_3}{K_3}=2,\] as desired. $\blacksquare$

1961 IMO (Problems) • Resources
Preceded by
Problem 3
1 2 3 4 5 6 Followed by
Problem 5
All IMO Problems and Solutions
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