Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of and meters, and there are no other distances. Give an example of how this could be.

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of m, Krash and Wally live on the same shore, m from each other. In summer, Dokko to Krash walk m, and Wally to Rosa - also m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa.

about Kikoriki/GoGoRiki / Smeshariki

Equal line segments are marked in triangle . Find its angles.

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points and are collinear.

Given a triangle , in which the angle is three times the angle . On the side , point is chosen such that the angle is twice the angle . Prove that .

Two turtles, the leader and the slave, are crawling along the plane from point to point . They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to . Prove that the slave turtle crawled no more than the leading one.

Cut a square into eight acute-angled triangles.

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in.

It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?

Cut an arbitrary triangle into pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a -gon and one is a -gon. Polygons do not have to be convex.

The altitudes and are drawn in triangle . Prove that the perpendicular bisector to the segment divides the segment in half.

Point is chosen inside triangle so that and On the side , a point is chosen such that . Prove that .

Angle bisectors and are drawn in triangle with angle . Find .

The segments connecting the interior point of a convex non-sided -gon with its vertices divide the -gon into congruent triangles. For what is the smallest that is possible?

Cut the grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of , the extensions of the other two sides also intersect and form an angle of . It is known that exactly one angle of the quadrilateral is . Find all of its other angles.

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?

In an acute-angled triangle on the side , point is chosen in such a way that . Points and are symmetric to with respect to vertices and , respectively. It turned out that . Find .

Inside the equilateral triangle , points and are chosen so that the quadrilateral is convex, and . Prove that .

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part.

Cut a square with three straight lines into three triangles and four quadrilaterals.

A quadrilateral is given, in which the lengths of some two sides are equal to and . Also, the diagonal of length divides it into two isosceles triangles. Find the perimeter of this quadrilateral.

Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are , and . Find the angles of the original triangle.

Fold the next seven corners into a rectangle.

Two equal rectangles of area are arranged as follows. Find the area of the gray rectangle.

A triangle is given in which . and . Find the length of the angle bisector drawn from the vertex , if it is known that the sides and differ by centimeters.

Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?

Let's call a corner the figure that is obtained by removing one cell from a square. Cut the square into corners so that no two of them form a or rectangle together.

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area?

The rectangle is cut into squares as shown in the figure on the right. Find its sides if the side of the smallest square is .

Inside the convex pentagon , a point was chosen, and it turned out that all five triangles , , , and are congrunet to each other. Prove that these triangles are isosceles or right-angled.

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

An isosceles triangle with base is given. On the rays , and , the points and were marked, respectively, in such a way that , and . Find the sum of the angles , and .

Squares and are located as shown in the figure. It turned out that points and lie on the same straight line. Prove that then the points and also lie on the same line.