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Some real numbers $ a_1, a_2, a_3,…,a _{1996} $ are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.

PFind the number of solutions in natural numbers of the equation [x / 10] = [x / 11] + 1.

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the k-th word is obtained from the $(k-1)$-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB …

$\\$ a) Where in this sequence will the 1000th letter “A” be?

$\\$ b) Prove that this sequence is non-periodic.

a$)$ Give an example of a positive number a such that {a} + {1 / a} = 1.

$\\$

b$)$ Can such an a be a rational number?

f$(x)$ is an increasing function defined on the interval [0, 1]. It is known that the range of its values belongs to the interval [0, 1]. Prove that, for any natural N, the graph of the function can be covered by N rectangles whose sides are parallel to the coordinate axes so that the area of each is $1/N^2$. $($In a rectangle we include its interior points and the points of its boundary$)$.

Prove that for every natural number n $>$ 1 the equality: [$n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

At the cat show, 10 male cats and 19 female cats sit in a row where next to each female cat sits a fatter male cat. Prove that next to each male cat is a female cat, which is thinner than it.

A schoolboy told his friend Bob:$\\$

“We have thirty-five people in the class. And imagine, each of them is friends with exactly eleven classmates …”$\\$

“It cannot be,” Bob, the winner of the mathematical Olympiad, answered immediately. Why did he decide this?

During the chess tournament, several players played an odd number of games. Prove that the number of such players is even.

The numbers a and b are such that the first equation of the system

$cos x = ax + b$

$sin x + a = 0$

has exactly two solutions. Prove that the system has at least one solution.

The numbers a and b are such that the first equation of the system

$sin x + a = bx$

$cos x = b$

has exactly two solutions. Prove that the system has at least one solution.

Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.

In a dark room on a shelf there are 4 pairs of socks of two different sizes and two different colours that are not arranged in pairs. What is the minimum number of socks necessary to move from the drawer to the suitcase, without leaving the room, so that there are two pairs of socks of different sizes and colours in the suitcase?

Is there a line on the coordinate plane relative to which the graph of the function $y = 2^x$ is symmetric?

Does there exist a number h such that for any natural number n the number [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

How can you connect 50 cities with the least number of airlines so that from every city you can get to any other one by making no more than two transfers?

A system of points connected by segments is called “connected” if from each point one can go to any other one along these segments. Is it possible to connect five points to a connected system so that when erasing any segment, exactly two connected points systems are formed that are not related to each other? $($We assume that in the intersection of the segments, the transition from one of them to another is impossible$)$.

n points are connected by segments so that each point is connected to something and there are no two points that would be connected in two different ways. Prove that the total number of segments is n – 1.

The numbers [a], [2a], …, [Na] are all different, and the numbers [1/a], [2/a],…, [M/a] are also all different. Find all such a.

The segment OA is given. From the end of the segment A there are 5 segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment $($except point O$)$.