Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 8-13.
Visit Beast Academy ‚
Books for Ages 8-13 Beast Academy Online
AoPS Academy
Nationwide learning centers
for students in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1966 IMO Problems/Problem 6"

(added solution)
(→‎Solution)
 
(One intermediate revision by one other user not shown)
Line 5: Line 5:
 
Let the lengths of sides <math>BC</math>, <math>CA</math>, and <math>AB</math> be <math>a</math>, <math>b</math>, and <math>c</math>, respectively. Let <math>BK=d</math>, <math>CL=e</math>, and <math>AM=f</math>.
 
Let the lengths of sides <math>BC</math>, <math>CA</math>, and <math>AB</math> be <math>a</math>, <math>b</math>, and <math>c</math>, respectively. Let <math>BK=d</math>, <math>CL=e</math>, and <math>AM=f</math>.
  
−
Now assume for the sake of contradiction that the areas of <math>\Delta AML</math>, <math>\Delta BKM</math>, and <math>\Delta CLK</math> are all at greater thanone fourth of that of <math>\Delta ABC</math>. Therefore
+
Now assume for the sake of contradiction that the areas of <math>\Delta AML</math>, <math>\Delta BKM</math>, and <math>\Delta CLK</math> are all at greater than one fourth of that of <math>\Delta ABC</math>. Therefore
  
 
<cmath>\frac{AM\cdot AL\sin{\angle BAC}}{2}>\frac{AB\cdot AC\sin{\angle BAC}}{8}</cmath>
 
<cmath>\frac{AM\cdot AL\sin{\angle BAC}}{2}>\frac{AB\cdot AC\sin{\angle BAC}}{8}</cmath>
Line 18: Line 18:
  
 
This is a contradiction, which shows that our assumption must have been false in the first place. This proves the desired result.
 
This is a contradiction, which shows that our assumption must have been false in the first place. This proves the desired result.
 +
 +
==Solution 2==
 +
Let <math>AR : AB = x, BP : BC = y, CQ : CA = z</math>. Then it is clear that the ratio of areas of <math>AQR, BPR, CPQ</math> to that of <math>ABC</math> equals <math>x(1-y), y(1-z), z(1-x)</math>, respectively. Suppose all three quantities exceed <math>\frac{1}{4}</math>. Then their product also exceeds <math>\frac{1}{64}</math>. However, it is clear by AM-GM that <math>x(1-x) \le \frac{1}{4}</math>, and so the product of all three quantities cannot exceed <math>\frac{1}{64}</math> (by the associative property of multiplication), a contradiction. Hence, at least one area is less than or equal to <math>\frac{1}{4} [ABC]</math>.
 +
 
== See Also ==
 
== See Also ==
 
{{IMO box|year=1966|num-b=5|after=Last Problem}}
 
{{IMO box|year=1966|num-b=5|after=Last Problem}}
 
[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]

Latest revision as of 00:37, 17 May 2015

Problem

In the interior of sides $BC, CA, AB$ of triangle $ABC$, any points $K, L,M$, respectively, are selected. Prove that the area of at least one of the triangles $AML, BKM, CLK$ is less than or equal to one quarter of the area of triangle $ABC$.

Solution

Let the lengths of sides $BC$, $CA$, and $AB$ be $a$, $b$, and $c$, respectively. Let $BK=d$, $CL=e$, and $AM=f$.

Now assume for the sake of contradiction that the areas of $\Delta AML$, $\Delta BKM$, and $\Delta CLK$ are all at greater than one fourth of that of $\Delta ABC$. Therefore

\[\frac{AM\cdot AL\sin{\angle BAC}}{2}>\frac{AB\cdot AC\sin{\angle BAC}}{8}\]

In other words, $AM\cdot AL>\frac{1}{4}AB\cdot AC$, or $f(b-e)>\frac{bc}{4}$. Similarly, $d(c-f)>\frac{ac}{4}$ and $e(a-d)>\frac{ab}{4}$. Multiplying these three inequalities together yields

\[def(a-d)(b-e)(c-f)>\frac{a^2b^2c^2}{64}\]

We also have that $d(a-d)\leq \frac{a^2}{4}$, $e(b-e)\leq \frac{b^2}{4}$, and $f(c-f)\leq \frac{c^2}{4}$ from the Arithmetic Mean-Geometric Mean Inequality. Multiplying these three inequalities together yields

\[def(a-d)(b-e)(c-f)\leq\frac{a^2b^2c^2}{64}\]

This is a contradiction, which shows that our assumption must have been false in the first place. This proves the desired result.

Solution 2

Let $AR : AB = x, BP : BC = y, CQ : CA = z$. Then it is clear that the ratio of areas of $AQR, BPR, CPQ$ to that of $ABC$ equals $x(1-y), y(1-z), z(1-x)$, respectively. Suppose all three quantities exceed $\frac{1}{4}$. Then their product also exceeds $\frac{1}{64}$. However, it is clear by AM-GM that $x(1-x) \le \frac{1}{4}$, and so the product of all three quantities cannot exceed $\frac{1}{64}$ (by the associative property of multiplication), a contradiction. Hence, at least one area is less than or equal to $\frac{1}{4} [ABC]$.

See Also

1966 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1966_IMO_Problems/Problem_6&oldid=70272"
Invalid username
Login to AoPS