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Difference between revisions of "1991 IMO Problems/Problem 1"

(New page: Given a triangle <math>ABC</math> let <math>I</math> be the center of its inscribed circle. The internal bisectors of the angles <math>A,B,C</math> meet the opposite sides in <math>A^\prim...)
 
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<math>\frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}</math>
 
<math>\frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}</math>
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== Solution ==
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We have <math>\prod\frac{AI}{AA^\prime}=\prod\frac{1}{1+\frac{IA^\prime}{IA}}</math>.  From Van Aubel's Theorem, we have <math>\frac{IA}{IA^\prime}=\frac{AB^\prime}{B^\prime C}+\frac{AC^\prime}{C^\prime B}</math> which from the Angle Bisector Theorem reduces to <math>\frac{b+c}{a}</math>.  We find similar expressions for the other terms in the product so that the product simplifies to <math>\prod\frac{1}{1+\frac{a}{b+c}}=\prod\frac{b+c}{a+b+c}</math>. Letting <math>a=x+y,b=y+z,c=z+x</math> for positive reals <math>x,y,z</math>, the product becomes <math>\frac{1}{8}\prod\frac{x+2y+z}{x+y+z}=\frac{1}{8}\prod\left(1+\frac{y}{x+y+z}\right)</math>.  To prove the right side of the inequality, we simply apply AM-GM to the product to get
 
We have <math>\prod\frac{AI}{AA^\prime}=\prod\frac{1}{1+\frac{IA^\prime}{IA}}</math>.  From Van Aubel's Theorem, we have <math>\frac{IA}{IA^\prime}=\frac{AB^\prime}{B^\prime C}+\frac{AC^\prime}{C^\prime B}</math> which from the Angle Bisector Theorem reduces to <math>\frac{b+c}{a}</math>.  We find similar expressions for the other terms in the product so that the product simplifies to <math>\prod\frac{1}{1+\frac{a}{b+c}}=\prod\frac{b+c}{a+b+c}</math>. Letting <math>a=x+y,b=y+z,c=z+x</math> for positive reals <math>x,y,z</math>, the product becomes <math>\frac{1}{8}\prod\frac{x+2y+z}{x+y+z}=\frac{1}{8}\prod\left(1+\frac{y}{x+y+z}\right)</math>.  To prove the right side of the inequality, we simply apply AM-GM to the product to get
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as desired.
 
as desired.
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Remark: To prove the right side of the inequality, a quicker way might be to use Gergonne's and AM-GM.
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== See Also == {{IMO box|year=1991|before=First Question|num-a=2}}

Latest revision as of 08:40, 19 November 2023

Given a triangle $ABC$ let $I$ be the center of its inscribed circle. The internal bisectors of the angles $A,B,C$ meet the opposite sides in $A^\prime,B^\prime,C^\prime$ respectively. Prove that

$\frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}$


Solution

We have $\prod\frac{AI}{AA^\prime}=\prod\frac{1}{1+\frac{IA^\prime}{IA}}$. From Van Aubel's Theorem, we have $\frac{IA}{IA^\prime}=\frac{AB^\prime}{B^\prime C}+\frac{AC^\prime}{C^\prime B}$ which from the Angle Bisector Theorem reduces to $\frac{b+c}{a}$. We find similar expressions for the other terms in the product so that the product simplifies to $\prod\frac{1}{1+\frac{a}{b+c}}=\prod\frac{b+c}{a+b+c}$. Letting $a=x+y,b=y+z,c=z+x$ for positive reals $x,y,z$, the product becomes $\frac{1}{8}\prod\frac{x+2y+z}{x+y+z}=\frac{1}{8}\prod\left(1+\frac{y}{x+y+z}\right)$. To prove the right side of the inequality, we simply apply AM-GM to the product to get

$\frac{1}{8}\prod\left(1+\frac{y}{x+y+z}\right)\le\frac{1}{8}\left(\frac{\sum1+\frac{y}{x+y+z}}{3}\right)^3=\frac{8}{27}$

To prove the left side of the inequality, simply multiply out the product to get

$\frac{1}{8}\left(1+\sum\frac{y}{x+y+z}+\sum\frac{yz}{(x+y+z)^2}+\frac{xyz}{(x+y+z)^3}\right)$

$>\frac{1}{8}\left(1+\sum\frac{y}{x+y+z}\right)=\frac{1}{4}$

as desired.

Remark: To prove the right side of the inequality, a quicker way might be to use Gergonne's and AM-GM.

See Also

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