In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.
120 unit squares are placed inside a \(20 \times 25\) rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.
You are given \(7\) straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than \(26^{\circ}\).
On a circle of radius 1, the point \(O\) is marked and from this point, to the right, a notch is marked using a compass of radius \(l\). From the obtained notch \(O_1\), a new notch is marked, in the same direction with the same radius and this is process is repeated 1968 times. After this, the circle is cut at all 1968 notches, and we get 1968 arcs. How many different lengths of arcs can this result in?
It is known that a camera located at \(O\) cannot see the objects \(A\) and \(B\), where the angle \(AOB\) is greater than \(179^\circ\). 1000 such cameras are placed in a Cartesian plane. All of the cameras simultaneously take a picture. Prove that there will be a picture taken in which no more than 998 cameras are visible.
In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.
In a square with side length 1 there is a broken line, which does not self-intersect, whose length is no less than 200. Prove that there is a straight line parallel to one of the sides of the square that intersects the broken line at a point no less than 101 units along the line.
A white plane is arbitrarily sprinkled with black ink. Prove that for any positive \(l\) there exists a line segment of length \(l\) with both ends of the same colour.
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
A page of a calendar is partially covered by the previous torn sheet (see the figure). The vertices A and B of the upper sheet lie on the sides of the bottom sheet. The fourth vertex of the lower leaf is not visible – it is covered by the top sheet. The upper and lower pages, of course, are identical in size to each other. Which part of the lower page is greater, that which is covered or that which is not?