Peter marks several cells on a $5 \times 5$ board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square $($you can only place the corners on the squares$)$. What is the smallest number of cells that Peter should mark so that Richard cannot win?
Cut an arbitrary triangle into 3 parts and out of these pieces construct a rectangle.
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius $\frac{1}{7}$.
A straight corridor of length 100m is covered with 20 rugs that have a total length of 1km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?
Prove that rational numbers from [0; 1] can be covered by a system of intervals of total length no greater than 1/1000.
Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside the quadrilateral.
10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.
Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than $360^\circ$.
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an 8×8 square grid, so that no more ‘corners’ would fit?
A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?
A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than 720$^{\circ}$.
The surface of a 3x3x3 Rubik’s Cube contains 54 squares. What is the maximum number of squares we can mark, so that no marked squares share a vertex or are directly adjacent to another marked square?
Kai has a piece of ice in the shape of a “corner” $($see the figure$)$. The Snow Queen demanded that Kai cut it into four equal parts. How can he do this?
At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L $($all tiles are oriented the same way$)$. Can he make a rectangle with a size of $5 \times 8$? $($The tiles can be rotated, but you cannot turn them over$)$. For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.
A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?
Cut the figure $($on the boundaries of cells$)$ into three equal parts $($the same in shape and size$)$.