Alice and Bob are playing a year number game.
There will be two game numbers and and one starting number from the set used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number is reached or exceeded.
Whoever reaches the number wins. If is exceeded, the game ends in a draw. Show that Bob cannot win. What starting number does Bob have to choose to prevent Alice from winning?
A square is given. Over the side draw an equilateral triangle on the outside. The midpoint of the segment is and the midpoint of the side is . Prove that .
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(Karl Czakler)
Source: Serbian selection contest for the IMO 2025
Let be an acute triangle. Let be the reflection of point over the line . Let and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment . Let and be the points where the reflection of line with respect to line intersects the circumcircle of triangle , where point lies on the arc not containing . If is a point on the line such that , prove that .
Let be a regular hexagon with sidelength s. The points and are on the diagonals and , respectively, such that . Prove that the three points , and are on a line.
Convex quadrilateral has . Point is the foot of the perpendicular from to . Points and lie on sides and , respectively, such that lies inside triangle and Prove that line is tangent to the circumcircle of triangle .