Kazakhstan 2012 ( Grade 10 problem 4)

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Kazakhstan 2012 ( Grade 10 problem 4)

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#1 h May 20, 2012, 5:38 PM • 2 Y

Y by Adventure10, Mango247

Let $k_{1},k_{2}, k_{3}$ -Excircles triangle $A_{1}A_{2}A_{3}$ with area $S$ . $k_{1}$ touch side $A_{2}A_{3}$ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$ .Let $S_{1}$ - area of the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$ . Prove that $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$

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#2 h May 23, 2012, 11:26 AM • 3 Y

Y by 125907, Adventure10, Mango247

Suppose, $a_1,a_2,a_3$ are the sides of $A{}_1A{}_2A{}_3$ and $s$ denotes the semi-perimeter of $\Delta A{}_1A{}_2A{}_3$ .
Then note that, $A{}_1B{}_1.A{}_1C{}_1=s^2$ . $\frac {S}{S_1}=\frac {A{}_1B{}_1}{A{}_1C{}_1}=\frac {A{}_1B{}_1^2}{s^2}$ . So it suffices to prove hat, $\sum A{}_1B{}_1^2\ge s^2$ .
By Stewart's theorem $a_1(A{}_1B{}_1^2+(s-a_2)(s{}_a{}_3))=a_2^2(s-a_3)+a_3^2(s-a_2)$ .
But note that, $a_2^2(s-a_3)+a_3^2(s-a_2)\ge a{}_1a{}_2a{}_3$ .
So $A{}_1B{}_1^2\ge s(a_2+a_3)-s^2$ .
So $\sum A{}_1B{}_1^2\ge s^2$ . So done.

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#3 h Dec 23, 2022, 10:06 PM • 3 Y

Y by Mango247, Mango247, Mango247

Let $k_{1},k_{2}, k_{3}$ be excircles of triangle $A_{1}A_{2}A_{3}$ with area $S$ . $k_{1}$ touch side $A_{2}A_{3}$ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$ .Let $S_{1}$ - area of the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$ . Prove that $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}.$

This post has been edited 2 times. Last edited by parmenides51, Jan 7, 2023, 11:58 PM

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