RU-SPB 2019

A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.

In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen?

Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

Call the improvement of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$ ), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$ . Prove that you can erase some of them, and improve each of the other numbers so that the sum the resulting numbers were again $2^{100}$ .

The bisectors $BB_1$ and $CC_1$ of the acute triangle $ABC$ intersect in point $I$ . On the extensions of the segments $BB_1$ and $CC_1$ , the points $B'$ and $C'$ are marked, respectively So, the quadrilateral $AB'IC'$ is a parallelogram. Prove that if $\angle BAC = 60^o$ , then the straight line $B'C'$ passes through the intersection point of the circumscribed circles of the triangles $BC_1B'$ and $CB_1C'$ .

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$ , and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.

Every two of the $n$ cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least $k$ flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest $k$ committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?

Given a convex quadrilateral $ABCD$ . The medians of the triangle $ABC$ intersect at point $M$ , and the medians of the triangle $ACD$ at point $N$ . The circle, circumscibed around the triangle $ACM$ , intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$ . Prove that $\angle AKD = \angle MKC$ .

A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$ ?

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$ .