Let , be the incenter and the -excenter of a triangle ; , be the touching points of the incircle with , respectively; be the common point of and . The perpendicular to from meets at point . Prove that , , , are concyclic.
Proposed by:Y.Shcherbatov

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)

As shown in the graph, an ant starts from and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching without passing through

Let be an acute triangle () with circumcircle and orthocenter . Let be the midpoint of . Line intersects again at , and line intersects again at . Points and lie on and respectively, such that is a parallelogram.

Prove that is tangent to the circumcircle of triangle .

Determine how many positive integers have the property that both
are consecutive perfect squares.

An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.
Proposed by: M.Evdokimov

Circles and are given. Let be the midpoint of the segment joining their centers, , be arbitrary points on , respectively such that . Find the locus of the midpoints of segments .
Proposed by: L Shatunov

Let , be the circumcenter and the incenter of an acute-angled scalene triangle ; , , be the touching points of its excircle with the side and the extensions of , respectively. Prove that if the orthocenter of the triangle lies on the circumcircle of , then it is symmetric to the midpoint of the arc with respect to .
Proposed by: P.Puchkov,E.Utkin

Let be a point inside a quadrilateral such that . Points , , are isogonally conjugated to with respect to the triangles , , , respectively. Prove that the diagonals of the quadrilaterals and concur.
Proposed by: G.Galyapin

Let be a quadrilateral such that the excircles and of triangles and touching their sides and respectively touch the extension of at the same point . The segment meets at point , and the line meets at and . Prove that one of angles and is right
Proposed by: I.Kukharchuk