Problems

Given a square trinomial \(f (x) = x^2 + ax + b\). It is known that for any real \(x\) there exists a real number \(y\) such that \(f (y) = f (x) + y\). Find the greatest possible value of \(a\).

In the infinite sequence \((x_n)\), the first term \(x_1\) is a rational number greater than 1, and \(x_{n + 1} = x_n + \frac{1}{\lfloor x_n\rfloor }\) for all positive integers \(n\).

Prove that there is an integer in this sequence.

Note that in this problem, square brackets represent integers and curly brackets represent non-integer values or 0.

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?

On the plane coordinate axes with the same but not stated scale and the graph of the function \(y = \sin x\), \(x\) \((0; \alpha)\) are given.

How can you construct a tangent to this graph at a given point using a compass and a ruler if: a) \(\alpha \in (\pi /2; \pi)\); b) \(\alpha \in (0; \pi /2)\)?

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.

In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?

The sequence \(a_1, a_2, \dots\) is such that \(a_1 \in (1,2)\) and \(a_{k + 1} = a_k + \frac{k}{a_k}\) for any positive integer \(k\). Prove that it cannot contain more than one pair of terms with an integer sum.The sequence \(a_1, a_2, \dots\) is such that \(a_1 \in (1,2)\) and \(a_{k + 1} = a_k + \frac{k}{a_k}\) for any positive integer \(k\). Prove that it cannot contain more than one pair of terms with an integer sum.

Prove that if the expression

takes a rational value, then the expression

also takes on a rational value.

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?

An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.