Problems
Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.
Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.
Let’s denote any two digits with the letters \(A\) and \(X\). Prove that the six-digit number \(XAXAXA\) is divisible by 7 without a remainder.
A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.
The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) intersect at two points, bounding a certain figure.
Find the equation of the vertical line dividing the area of this figure in half.
100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.
In 25 boxes there are spheres of different colours. It is known that for any \(k\) where \(1 \leq k \leq 25\) in any \(k\) of the boxes there are spheres of exactly \(k+1\) different colours. Prove that a sphere of one particular colour lies in every single box.
When water is drained from a pool, the water level \(h\) in it varies depending on the time \(t\) according to the function \(h (t) = at^2 + bt + c\), and at the time \(t_0\) of when the draining is ending, the equalities \(h (t_0) = h' (t_0) = 0\) are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?
A game of ’Battleships’ has a fleet consisting of one \(1\times 4\) square, two \(1\times 3\) squares, three \(1\times 2\) squares, and four \(1\times 1\) squares. It is easy to distribute the fleet of ships on a \(10\times 10\) board, see the example below. What is the smallest square board on which this fleet can be placed? Note that by the rules of the game, no two ships can be placed on horizontally, vertically, or diagonally adjacent squares.
On an 8×8 grid (like a chessboard), an L-corner is a shape made of 3 little squares of the board that touch to make an L. You can turn the L any way you like. We place the L-corners so that none overlap. What is the fewest L-corners you must place so that no more L-corners can be added anywhere? Here is an example of how three L-corners may look like:
