Olympiad problems

2024 Korea - Final Round, P4

For a triangle $ABC$, $O$ is the circumcircle and $D$ is a point on ray $BA$. $E$ and $F$ are points on $O$ so that $DE$ and $DF$ are tangent to $O$ and $EF$ cuts $AC$ at $T(\neq C)$. $P(\neq B,C)$ is a point on the arc $BC$ not containing $A$, and $DP$ cuts $O$ at $Q (\neq P)$. Let $BQ$ and $DT$ meets on $X (\neq Q)$, and $PT$ cuts $O$ at $Y (\neq P)$. Prove that $C,X,Y$ are collinear.

2013 Saudi Arabia IMO TST, 1

Tags: concurrency , incircle , desargue , pascal , tangent , geometry , collinear

Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.

1998 VJIMC, Problem 4-I

Tags: pascal

Prove that there exists a program in standard Pascal which prints out its own ASCII code. No disk operations are permitted.

2013 Danube Mathematical Competition, 1

Tags: pascal , concurrency , geometry

Given six points on a circle, $A, a, B, b, C, c$, show that the Pascal lines of the hexagrams $AaBbCc, AbBcCa, AcBaCb$ are concurrent.