Let be an acute-angled, non-isosceles triangle with circumcenter and incenter , such that


where , , and .

Let , , and be the excenters opposite to vertices , , and , respectively, and let , , and be the Lemoine points of triangles , , and , respectively.

Prove that the circles , , and all pass through a common point . Moreover, the isogonal conjugate of with respect to lies on the line .

Note (Hint)

Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?

The diagonals of a convex quadrilateral intersect at point . The points of tangency of the circumcircles of triangles and with their common external tangents lie on a circle . The points of tangency of the circumcircles of triangles and with their common external tangents lie on a circle . Prove that the centers of circles and coincide.

Petya and Vasya are playing a game on an initially empty grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?

Inside triangle , point is marked. Point is on segment , and point is on segment such that the circumcircles of triangles and are tangent to line . Lines are drawn through points and passing through the center of the circumcircle of triangle , and through points and passing through the center of the circumcircle of triangle . Prove that there exists a circle tangent to all four drawn lines.

Find all natural numbers for which there exists an even natural number such that the number
is a perfect square.

In the plane, points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than . For every group of marked points, he calculated the sum of the numbers on all connecting segments. It turned out that the absolute value of each such sum is at least , and there are both positive and negative such sums. What is the maximum possible value of ?

pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers and from the board such that:
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers and from the board such that:

A right prism is given. It is known that triangles , , , and are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle , lie on the same sphere.

A pair of polynomials and with integer coefficients is called if from the divisibility of both differences and by , it follows that both and are divisible by 100. Does there exist such an important pair of polynomials , , such that the pair and is also important?