Found problems: 4
2023 Turkey MO (2nd round), 2
Let $ABC$ be a triangle and $P$ be an interior point. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally and let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1$, $B_2$, $C_1$, $C_2$ analogously. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ and $OP$ are concurrent.
2008 Dutch Mathematical Olympiad, 4
Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent.
In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles.
Find the radius of $C_4$.
[asy]
unitsize(0.4 cm);
pair[] O;
real[] r;
O[1] = (-12/5,16/5);
r[1] = 1;
O[2] = (0,5);
r[2] = 2;
O[3] = (0,0);
r[3] = 3;
O[4] = (-132/115, 351/115);
r[4] = 6/23;
draw(Circle(O[1],r[1]));
draw(Circle(O[2],r[2]));
draw(Circle(O[3],r[3]));
draw(Circle(O[4],r[4]));
label("$C_1$", O[1]);
label("$C_2$", O[2]);
label("$C_3$", O[3]);
[/asy]
2019 Brazil Team Selection Test, 2
Let $ABC$ be a triangle, and $A_1$, $B_1$, $C_1$ points on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $A_1B_1C_1$ is equilateral. Let $I_1$ and $\omega_1$ be the incenter and the incircle of $AB_1C_1$. Define $I_2$, $\omega_2$ and $I_3$, $\omega_3$ similarly, with respect to the triangles $BA_1C_1$ and $CA_1B_1$, respectively. Let $l_1 \neq BC$ be the external tangent line to $\omega_2$ and $\omega_3$. Define $l_2$ and $l_3$ similarly, with respect to the pairs $\omega_1$, $\omega_3$ and $\omega_1$, $\omega_2$.
Knowing that $A_1I_2 = A_1I_3$, show that the lines $l_1$, $l_2$, $l_3$ are concurrent.
2001 Bosnia and Herzegovina Team Selection Test, 4
In plane there are two circles with radiuses $r_1$ and $r_2$, one outside the other. There are two external common tangents on those circles and one internal common tangent. The internal one intersects external ones in points $A$ and $B$ and touches one of the circles in point $C$. Prove that
$AC \cdot BC=r_1\cdot r_2$