Problems

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.

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) \(\alpha \in (0; \pi /2)\)?

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.

The nonzero numbers \(a\), \(b\), \(c\) are such that every two of the three equations \(ax^{11} + bx^4 + c = 0\), \(bx^{11} + cx^4 + a = 0\), \(cx^{11} + ax^4 + b = 0\) have a common root. Prove that all three equations have a common root.

The teacher wrote on the board in alphabetical order all possible \(2^n\) words consisting of \(n\) letters A or B. Then he replaced each word with a product of \(n\) factors, correcting each letter A by \(x\), and each letter B by \((1 - x)\), and added several of the first of these polynomials in \(x\). Prove that the resulting polynomial is either a constant or increasing function in \(x\) on the interval \([0, 1]\).

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

12 straight lines passing through the origin are drawn on a plane. Prove that it is possible to choose two of these lines such that the angle between them is less than 17 degrees.

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?

There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.