SHARYGIN 2008

(A.Zaslavsky, 9--10) The reflections of diagonal $BD$ of a quadrilateral $ABCD$ in the bisectors of angles $B$ and $D$ pass through the midpoint of diagonal $AC$ . Prove that the reflections of diagonal $AC$ in the bisectors of angles $A$ and $C$ pass through the midpoint of diagonal $BD$ (There was an error in published condition of this problem).

(A.Zaslavsky, 9--10) Quadrilateral $ABCD$ is circumscribed arounda circle with center $I$ . Prove that the projections of points $B$ and $D$ to the lines $IA$ and $IC$ lie on a single circle.

(A.Zaslavsky, 9--10) Given four points $A$ , $B$ , $C$ , $D$ . Any two circles such that one of them contains $A$ and $B$ , and the other one contains $C$ and $D$ , meet. Prove that common chords of all these pairs of circles pass through a fixed point.

(A.Myakishev, 9--10) Given a triangle $ABC$ . Point $A_1$ is chosen on the ray $BA$ so that segments $BA_1$ and $BC$ are equal. Point $A_2$ is chosen on the ray $CA$ so that segments $CA_2$ and $BC$ are equal. Points $B_1$ , $B_2$ and $C_1$ , $C_2$ are chosen similarly. Prove that lines $A_1A_2$ , $B_1B_2$ , $C_1C_2$ are parallel.

(A.Myakishev, 9--10) Given triangle $ABC$ . One of its excircles is tangent to the side $BC$ at point $A_1$ and to the extensions of two other sides. Another excircle is tangent to side $AC$ at point $B_1$ . Segments $AA_1$ and $BB_1$ meet at point $N$ . Point $P$ is chosen on the ray $AA_1$ so that $AP=NA_1$ . Prove that $P$ lies on the incircle.

(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this angle (There was an error in published condition of this problem).

(M.Volchkevich, 9--11) Given two circles and point $P$ not lying on them. Draw a line through $P$ which cuts chords of equal length from these circles.

(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points $A$ and $B$ . Points $X$ , $Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $AX$ and $BY$ .

(A.Myakishev, 9--11) Given triangle $ABC$ and a ruler with two marked intervals equal to $AC$ and $BC$ . By this ruler only, find the incenter of the triangle formed by medial lines of triangle $ABC$ .

(A.Abdullayev, 9--11) Prove that the triangle having sides $a$ , $b$ , $c$ and area $S$ satisfies the inequality $$a^2+b^2+c^2-\frac12(|a-b|+|b-c|+|c-a|)^2\geq 4\sqrt3 S.$$

(V.Protasov, 10-11) Given parallelogram $ABCD$ such that $AB = a$ , $AD = b$ . The first circle has its center at vertex $A$ and passes through $D$ , and the second circle has its center at $C$ and passes through $D$ . A circle with center $B$ meets the first circle at points $M_1$ , $N_1$ , and the second circle at points $M_2$ , $N_2$ . Determine the ratio $M_1N_1/M_2N_2$ .

(A.Zaslavsky, 10--11) a) Some polygon has the following property: if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric? b) Is it true that any figure with the property from part a) is central symmetric?

(A.Zaslavsky, B.Frenkin, 10--11) In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle.

a) All three segments are equal. Is it true that the triangle is equilateral?

b) Two segments are equal. Is it true that the triangle is isosceles?

c) Can the segments have length 4, 4 and 3?

(A.Khachaturyan, 10--11) a) All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?

b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?

(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $A$ belonging to both spheres. Prove that there is a point $B$ in the space with the following property: if an arbitrary circle passes through points $A$ and $B$ then the second points of its meet with the given spheres are equidistant from $B$ .

(I.Bogdanov, 11) Let $h$ be the least altitude of a tetrahedron, and $d$ the least distance between its opposite edges. For what values of $t$ the inequality $d>th$ is possible?