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Source: Macedonia National Olympiad 2014

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#1 h Apr 13, 2014, 10:23 AM • 3 Y

Y by StefanS, Adventure10, Mango247

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$ , circles $k_1$ and $k_3$ intersect and $C$ and $P$ , circles $k_2$ and $k_3$ intersect at $B$ and $P$ . Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$ , such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$ .

a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$
b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$ . Is it possible to reach equation? $*Note:$ P$ denotes the area of a triangle*

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#2 h Oct 27, 2014, 6:53 PM • 3 Y

Y by StefanS, Adventure10, Mango247

$a)$ $\angle AYB$ and $\angle APB$ are supplement, so are $\angle O_1O_2O_3$ and $\angle APB$ . Thus $\angle XYZ =\angle O_1O_2O_3$ . Similarly, $\angle YXZ =\angle O_2O_1O_3$ and $\angle XZY =\angle O_1O_3O_2$ , and we're done with $a)$ .

$b)$ Draw a line through $A$ and parallel $O_1O_2$ to intersect $k_1$ and $k_2$ at $D$ and $E$ ; the line $BE$ which parallels $O_2O_3$ intersects $k_3$ at $F$ . Triangles $DEF$ and $XYZ$ are similar because of equal angles. However, $XY \le DE$ (I let you prove this) and $[DEF]$ = $4$ × $[O_1O_2O_3]$ and we're done with $b)$ . Equation $P_{\triangle XYZ} = 4P_{\triangle O_1O_2O_3}$ is reached (sic) when $XYZ$ is $DEF$ .

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#3 h Dec 11, 2022, 9:12 AM

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Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$ , circles $k_1$ and $k_3$ intersect and $C$ and $P$ , circles $k_2$ and $k_3$ intersect at $B$ and $P$ . Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$ , such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$ .

a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$
b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$ . Is it possible to reach equation?

Note: $P$ denotes the area of a triangle.

This post has been edited 1 time. Last edited by parmenides51, Dec 12, 2022, 9:09 PM

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