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Difference between revisions of "1982 USAMO Problems/Problem 3"

(Created page with "== Problem == If a point <math>A_1</math> is in the interior of an equilateral triangle <math>ABC</math> and point <math>A_2</math> is in the interior of <math>\triangle{A_1BC}</...")
 
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== Solution ==
 
== Solution ==
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{{solution}}
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First, an arbitrary triangle <math>ABC</math> has isoperimetric quotient (using the notation <math>[ABC]</math> for area and <math>s = \frac{a + b + c}{2}</math>):
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<cmath>\frac{[ABC]}{4s^2} = \frac{[ABC]^3}{4s^2 [ABC]^2} = \frac{r^3 s^3}{4s^2 \cdot s(s-a)(s-b)(s-c)} = \frac{r^3}{4(s-a)(s-b)(s-c)}</cmath>
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<cmath>= \frac{1}{4} \cdot \frac{r}{s-a} \cdot \frac{r}{s-b} \cdot \frac{r}{s-c} = \frac{1}{4} \tan A/2 \tan B/2 \tan C/2.</cmath>
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Lemma. <math>\tan x \tan (A - x)</math> is increasing on <math>0 < x < \frac{A}{2}</math>, where <math>0 < A < 90^\circ</math>.
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Proof. <cmath>\tan x \tan (A - x) = \tan x \cdot \frac{\tan A - \tan x}{1 + \tan A \tan x} = 1 - \frac{1}{\cos^2 x (1 + \tan A \tan x)}</cmath>
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<cmath>= 1 - \frac{2}{1 + \cos 2x + \tan A \sin 2x} = 1 - \frac{2}{1 + \sec A \cos (A - 2x)}</cmath> is increasing on the desired interval, because <math>\cos (A - 2x)</math> is increasing on <math>0 < x < \frac{A}{2}.</math>
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Let <math>x_1, y_1, z_1</math> and <math>x_2, y_2, z_2</math> be half of the angles of triangles <math>A_1 BC</math> and <math>A_2 BC</math> in that order, respectively. Then it is immediate that <math>30^\circ > y_1 > y_2</math>, <math>30^\circ > z_1 > z_2</math>, and <math>x_1 + y_1 + z_1 = x_2 + y_2 + z_2 = 90^\circ</math>. Hence, by Lemma it follows that
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<cmath>\tan x_1 \tan y_1 \tan z_1 = \tan (90^\circ - y_1 - z_1) \tan y_1 \tan z_1 > \tan (90^\circ - y_1 - z_2) \tan y_1 \tan z_2</cmath>
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<cmath> > \tan (90^\circ - y_2 - z_2) \tan y_2 \tan z_2 = \tan x_2 \tan y_2 \tan z_2.</cmath>
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Multiplying this inequality by <math>\frac{1}{4}</math> gives that <math>I.Q[A_1 BC] > I.Q[A_2 BC]</math>, as desired.
  
 
== See Also ==
 
== See Also ==
 
{{USAMO box|year=1982|num-b=2|num-a=4}}
 
{{USAMO box|year=1982|num-b=2|num-a=4}}
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{{MAA Notice}}
  
 
[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]

Latest revision as of 22:29, 18 May 2015

Problem

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that

$I.Q. (A_1BC) > I.Q.(A_2BC)$,

where the isoperimetric quotient of a figure $F$ is defined by

$I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}$

Solution

First, an arbitrary triangle $ABC$ has isoperimetric quotient (using the notation $[ABC]$ for area and $s = \frac{a + b + c}{2}$): \[\frac{[ABC]}{4s^2} = \frac{[ABC]^3}{4s^2 [ABC]^2} = \frac{r^3 s^3}{4s^2 \cdot s(s-a)(s-b)(s-c)} = \frac{r^3}{4(s-a)(s-b)(s-c)}\] \[= \frac{1}{4} \cdot \frac{r}{s-a} \cdot \frac{r}{s-b} \cdot \frac{r}{s-c} = \frac{1}{4} \tan A/2 \tan B/2 \tan C/2.\]

Lemma. $\tan x \tan (A - x)$ is increasing on $0 < x < \frac{A}{2}$, where $0 < A < 90^\circ$.

Proof. \[\tan x \tan (A - x) = \tan x \cdot \frac{\tan A - \tan x}{1 + \tan A \tan x} = 1 - \frac{1}{\cos^2 x (1 + \tan A \tan x)}\] \[= 1 - \frac{2}{1 + \cos 2x + \tan A \sin 2x} = 1 - \frac{2}{1 + \sec A \cos (A - 2x)}\] is increasing on the desired interval, because $\cos (A - 2x)$ is increasing on $0 < x < \frac{A}{2}.$

Let $x_1, y_1, z_1$ and $x_2, y_2, z_2$ be half of the angles of triangles $A_1 BC$ and $A_2 BC$ in that order, respectively. Then it is immediate that $30^\circ > y_1 > y_2$, $30^\circ > z_1 > z_2$, and $x_1 + y_1 + z_1 = x_2 + y_2 + z_2 = 90^\circ$. Hence, by Lemma it follows that \[\tan x_1 \tan y_1 \tan z_1 = \tan (90^\circ - y_1 - z_1) \tan y_1 \tan z_1 > \tan (90^\circ - y_1 - z_2) \tan y_1 \tan z_2\] \[> \tan (90^\circ - y_2 - z_2) \tan y_2 \tan z_2 = \tan x_2 \tan y_2 \tan z_2.\] Multiplying this inequality by $\frac{1}{4}$ gives that $I.Q[A_1 BC] > I.Q[A_2 BC]$, as desired.

See Also

1982 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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