Olympiad problems

2019 India Regional Mathematical Olympiad, 2

Tags: p2 , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.

2010 Contests, 2

Tags: inversion , p2 , geometry

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

2010 IMO, 2

Tags: inversion , p2 , geometry

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

2010 IMO Shortlist, 4

Tags: geometry , inversion , p2

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

2020 DMO Stage 1, 2.

Tags: combinatorics , dmo day 2 , p2

[b]Q[/b] On a $10 \times 10$ chess board whose colors of square are green and blue in an arbitrary way and we are simultaneously allowed to switch all the colors of all squares in any $(2 \times 2)$ and $(5\times 5)$ region. Can we transform any coloring of the board into one where all squares are blue ? Give a proper explanation of your answer. Note. that if a unit square is part of both the $2\times 2$ and $5\times 5$ region,then its color switched is twice(i.e switching is additive) [i]Proposed by Aritra12[/i]