Olympiad problems

2025 Macedonian Balkan MO TST, 1

A set of $n \ge 2$ light bulbs are arranged around a circle, and are consecutively numbered with $1, 2, . . . , n$. Each bulb can be in one of two states: either it is [b]on[/b] or [b]off[/b]. In the initial configuration, at least one bulb is turned on. On each one of $n$ days we change the current on/off configuration as follows: for $1 \le k \le n$, on the $k$-th day we start from the $k$-th bulb and moving in clockwise direction along the circle, we change the state of every traversed bulb until we switch on a bulb which was previously off. Prove that the final configuration, reached on the $n$-th day, coincides with the initial one.

2022 Olympic Revenge, Problem 2

Tags: geometry , configuration

Let $ABC$ be a triangle and $\Omega$ its circumcircle. Let the internal angle bisectors of $\angle BAC, \angle ABC, \angle BCA$ intersect $BC,CA,AB$ on $D,E,F$, respectively. The perpedincular line to $EF$ through $D$ intersects $EF$ on $X$ and $AD$ intersects $EF$ on $Z$. The circle internally tangent to $\Omega$ and tangent to $AB,AC$ touches $\Omega$ on $Y$. Prove that $(XYZ)$ is tangent to $\Omega$.

KoMaL A Problems 2021/2022, A. 806

Tags: geometry , komal , configuration

Four distinct lines are given in the plane, which are not concurrent and no three of which are parallel. Prove that it is possible to find four points in the plane, $A,B,C,$ and $D$ with the following properties: [list=1] [*]$A,B,C,$ and $D$ are collinear in this order; [*]$AB=BC=CD$; [*]with an appropriate order of the four given lines, $A$ is on the first, $B$ is on the second, $C$ is on the third and $D$ is on the fourth line. [/list] [i]Proposed by Kada Williams, Cambridge[/i]