Given , let
and

Prove that .

Given triangle inscribed in , is the midpoint of arc of . The perpendicular bisector intersects at . intersects at different from . is the orthocenter of triangle . Prove that =

A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules:
i) If the frog jumps one square, it then turns to the right;
ii) If the frog jumps two squares, it then turns to the left.

Is it possible for the frog to reach the square exactly squares north of the initial square after some finite number of moves if it is initially facing:
a) North;
b) East?

Merlin's castle has 100 rooms and 1000 corridors. Each corridor links some two rooms. Each pair of rooms is linked by one corridor at most. Merlin has given out the plan of the castle to the wise men and declared the rules of the challenge. The wise men need to scatter across the rooms in a manner they wish. Each minute Merlin will choose a corridor and one of the wise men will have to pass along it from one of the rooms at its ends to the other one. Merlin wins when in both rooms on the ends of the chosen corridor there are no wise men. Let us call a number the magic number of the castle if wise men can pre-agree before the challenge and act in such a way that Merlin never wins, being the minimal possible number. What are the possible values of the magic number of the castle? (Merlin and all the wise men always know the location of all the wise men).

Timofey Vasilyev

Let be a cyclic quadrilateral, and let , , , and be the midpoints of , , , and respectively. Let , , and be the orthocenters of triangles , , and , respectively. Prove that the quadrilaterals and have the same area.