The surface of a \(3\times 3\times 3\) 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?
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 \(720^{\circ}\).
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?
What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an \(8\times 8\) square grid, so that no more ’corners’ would fit?
Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
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 \(360^\circ\).
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.
Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
A straight corridor of length 100 m is covered with 20 rugs that have a total length of 1 km. 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?