SHARYGIN 2006

Two straight lines intersecting at an angle of $46^o$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?

Points $A, B$ move with equal speeds along two equal circles. Prove that the perpendicular bisector of $AB$ passes through a fixed point.

The map shows sections of three straight roads connecting the three villages, but the villages themselves are located outside the map. In addition, the fire station located at an equal distance from the three villages is not indicated on the map, although its location is within the map. Is it possible to find this place with the help of a compass and a ruler, if the construction is carried out only within the map?

a) Given two squares $ABCD$ and $DEFG$ , with point $E$ lying on the segment $CD$ , and points $F,G$ outside the square $ABCD$ . Find the angle between lines $AE$ and $BF$ . b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$ , and the points $R, S, T$ are outside the pentagon $OKLMN$ . Find the angle between straight lines $KP$ and $MS$ .

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$ ). The strip can be bent, but not torn.

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$ . Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$ . b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$ . Draw through $A$ and $B$ a circle tangent to a circle of radius $r$ .

The point $E$ is taken inside the square $ABCD$ , the point $F$ is taken outside, so that the triangles $ABE$ and $BCF$ are congruent . Find the angles of the triangle $ABE$ , if it is known that $EF$ is equal to the side of the square, and the angle $BFD$ is right.

The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$ . Find the length of $AB$ .

$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$ . Prove that if $a + b + c = 2\pi$ , then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.

At what $n$ can a regular $n$ -gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$ . Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.

In the triangle $ABC$ , the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$ . Prove that if $\angle A$ is obtuse, then $AB = AC$ .

Two straight lines $a$ and $b$ are given and also points $A$ and $B$ . Point $X$ slides along the line $a$ , and point $Y$ slides along the line $b$ , so that $AX \parallel BY$ . Find the locus of the intersection point of $AY$ with $XB$ .

Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$ , where $AB$ is the diameter of the circle.

A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$ , respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$ , and $M$ is the midpoint of the segment $PA_1$ . Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.

Regular triangles are built on the sides of the triangle $ABC$ . It turned out that their vertices form a regular triangle. Is the original triangle regular also?

In two circles intersecting at points $A$ and $B$ , parallel chords $A_1B_1$ and $A_2B_2$ are drawn. The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$ . Prove that $XY // A_1B_1$ .

Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$ , one of which intersects $BC$ at point $X$ , and the other intersects $AC$ at point $Y$ . Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$ . Prove that the points $X, Y, Z$ lie on the same line.

Through the midpoints of the sides of the triangle $T$ , straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$ . Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$ .

Four points are given $A, B, C, D$ . Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD$ , $CDA$ , $DAB$ , $ABC$ and $A_2,$ $B_2$ , $C_2$ , $D_2$ are orthocenters of the triangles $B_1C_1D_1$ , $C_1D_1A_1$ , $D_1A_1B_1$ , $A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.

On the sides $AB, BC, CA$ of triangle $ABC$ , points $C', A', B'$ are taken. Prove that for the areas of the corresponding triangles, the inequality holds: $$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$ and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$ . Find the locus of the centers of the circles circumscribed around the triangles $PXY$ .

$ABCD$ is a convex quadrangle, $G$ is its center of gravity as a homogeneous plate (i.e., the intersection point of two lines, each of which connects the centroids of triangles having a common diagonal). a) Suppose that around $ABCD$ we can circumscribe a circle centered on $O$ . We define $H$ similarly to $G$ , taking orthocenters instead of centroids. Then the points of $H, G, O$ lie on the same line and $HG: GO = 2: 1$ . b) Suppose that in $ABCD$ we can inscribe a circle centered on $I$ . The Nagel point N of the circumscribed quadrangle is the intersection point of two lines, each of which passes through points on opposite sides of the quadrangle that are symmetric to the tangent points of the inscribed circle relative to the midpoints of the sides. (These lines divide the perimeter of the quadrangle in half). Then $N, G, I$ lie on one straight line, with $NG: GI = 2: 1$ .

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$ . Find the geometric locus of the projections of $P$ on the lines $AB$ . b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$ . Find the geometrical locus of the projections $P$ on the $ABC$ plane

In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$ , and $DA$ edges are equal to $\alpha$ , and for the remaining edges equal to $\beta$ . Find the ratio $AB / CD$ .

Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

Inscribe the equilateral triangle of the largest perimeter in a given semicircle.

What $n$ is the smallest such that "there is a $n$ -gon that can be cut into a triangle, a quadrilateral, ..., a $2006$ -gon''?

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$ . The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$ .

Two equal circles intersect at points $A$ and $B$ . $P$ is the point of one of the circles that is different from $A$ and $B, X$ and $Y$ are the second intersection points of the lines of $PA, PB$ with the other circle. Prove that the line passing through $P$ and perpendicular to $AB$ divides one of the arcs $XY$ in half.

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$ . Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$ .

Given a circle of radius $K$ . Two other circles, the sum of the radii of which are also equal to $K$ , tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$ . Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$ , a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$ . We similarly construct $B'$ and $C'$ . Prove that $A', B',C'$ lie on one straight line.

In a non-convex hexagon, each angle is either $90$ or $270$ degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?

A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

A convex quadrilateral $ABC$ is given. $A',B',C',D'$ are the orthocenters of triangles $BCD, CDA, DAB, ABC$ respectively. Prove that in the quadrilaterals $ABCP$ and $A'B'C'D'$ , the corresponding diagonals share the intersection points in the same ratio.