Petya and Vasya only know positive integers not exceeding $10^9-4000$. Petya considers numbers as good which are representable in the form $abc+ab+ac+bc$, where $a,b$ and $c$ are natural numbers not less than $100$. Vasya considers numbers as good which are representable in the form $xyz-x-y-z$, where $x,y$ and $z$ are natural numbers strictly bigger than $100$. For which of them are there more good numbers? Proposed by I. Bogdanov

A positive integer has exactly $50$ divisors. Is it possible that no difference of two different divisors is divisible by $100$? Proposed by A. Chironov

Two boys are given a bag of potatoes, each bag containing $150$ tubers. They take turns transferring the potatoes, where in each turn they transfer a non-zero tubers from their bag to the other boy's bag. Their moves must satisfy the following condition: In each move, a boy must move more tubers than he had in his bag before any of his previous moves (if there were such moves). So, with his first move, a boy can move any non-zero quantity, and with his fifth move, a boy can move $200$ tubers, if before his first, second, third and fourth move, the numbers of tubers in his bag was less than $200$. What is the maximal total number of moves the two boys can do? Proposed by E. Molchanov

In cyclic quadrilateral $ABCD$, $\angle A+ \angle D=\frac{\pi}{2}$. $AC$ intersects $BD$ at ${E}$. A line ${l}$ cuts segment $AB, CD, AE, DE$ at $X, Y, Z, T$ respectively. If $AZ=CE$ and $BE=DT$, prove that the diameter of the circumcircle of $\triangle EZT$ equals $XY$.

A neighborhood consists of $10 \times 10$ squares. On New Year's Eve it snowed for the first time and since then exactly $10$ cm of snow fell on each square every night (and snow fell only at night). Every morning, the janitor selects one row or column and shovels all the snow from there onto one of the adjacent rows or columns (from each cell to the adjacent side). For example, he can select the seventh column and from each of its cells shovel all the snow into the cell of the left of it. You cannot shovel snow outside the neighborhood. On the evening of the 100th day of the year, an inspector will come to the city and find the cell with the snowdrift of maximal height. The goal of the janitor is to ensure that this height is minimal. What height of snowdrift will the inspector find? Proposed by A. Solynin

The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$. Proposed by A. Tereshin

There are $8$ different quadratic trinomials written on the board, among them there are no two that add up to a zero polynomial. It turns out that if we choose any two trinomials $g_1(x), g_2(X)$ from the board, then the remaining $6$ trinomials can be denoted as $g_3(x),g_4(x),\dots,g_8(x)$ so that all four polynomials $g_1(x)+g_2(x),g_3(x)+g_4(x),g_5(x)+g_6(x)$ and $g_7(x)+g_8(x)$ have a common root. Do all trinomials on the board necessarily have a common root? Proposed by S. Berlov

$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.) Proposed by I. Bogdanov

Let $p$ and $q$ be different prime numbers. We are given an infinite decreasing arithmetic progression in which each of the numbers $p^{23}, p^{24}, q^{23}$ and $q^{24}$ occurs. Show that the numbers $p$ and $q$ also occur in this progression. Proposed by A. Kuznetsov

Let $n \ge 3$ be an odd integer. In a $2n \times 2n$ board, we colour $2(n-1)^2$ cells. What is the largest number of three-square corners that can surely be cut out of the uncoloured figure? Proposed by G. Sharafetdinova

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? Proposed by L. Shatunov

Let $ABCD$ be a convex quadrilateral with $\angle A+\angle D=90^\circ$ and $E$ the point of intersection of its diagonals. The line $\ell$ cuts the segments $AB$, $CD$, $AE$ and $ED$ in points $X,Y,Z,T$, respectively. Suppose that $AZ=CE$ and $BE=DT$. Prove that the length of the segment $XY$ is not larger than the diameter of the the circumcircle of $ETZ$. Proposed by A. Kuznetsov, I. Frolov

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. Proposed by T. Korotchenko

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. Proposed by A. Tereshin

Let $x_1$ and $x_2$ be positive integers. On a straight line, $y_1$ white segments and $y_2$ black segments are given, with $y_1 \ge x_1$ and $y_2 \ge x_2$. Suppose that no two segments of the same colour intersect (and do not have common ends). Moreover, suppose that for any choice of $x_1$ white segments and $x_2$ black segments, some pair of selected segments will intersect. Prove that $(y_1-x_1)(y_2-x_2)<x_1x_2$. Proposed by G. Chelnokov

Let $n>2$ be a positive integer. Masha writes down $n$ natural numbers along a circle. Next, Taya performs the following operation: Between any two adjacent numbers $a$ and $b$, she writes a divisor of the number $a+b$ greater than $1$, then Taya erases the original numbers and obtains a new set of $n$ numbers along the circle. Can Taya always perform these operations in such a way that after some number of operations, all the numbers are equal? Proposed by T. Korotchenko

We are given an infinite cylinder in space (i.e. the locus of points of a given distance $R>0$ from a given straight line). Can six straight lines containing the edges of a tetrahedron all have exactly one common point with this cylinder? Proposed by A. Kuznetsov

Call a triple $(a,b,c)$ of positive numbers mysterious if \[\sqrt{a^2+\frac{1}{a^2c^2}+2ab}+\sqrt{b^2+\frac{1}{b^2a^2}+2bc}+\sqrt{c^2+\frac{1}{c^2b^2}+2ca}=2(a+b+c).\]Prove that if the triple $(a,b,c)$ is mysterious, then so is the triple $(c,b,a)$. Proposed by A. Kuznetsov, K. Sukhov

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? Proposed by I. Bogdanov, K. Knop

A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\omega$. We draw a line $\ell_a \parallel BC$ through the point $A$, a line $\ell_b \parallel CD$ through the point $B$, a line $\ell_c \parallel DA$ through the point $C$, and a line $\ell_d \parallel AB$ through the point $D$. Suppose that the quadrilateral whose successive sides lie on these four straight lines is inscribed in a circle $\gamma$ and that $\omega$ and $\gamma$ intersect in points $E$ and $F$. Show that the lines $AC, BD$ and $EF$ intersect in one point. Proposed by A. Kuznetsov

Let $ABC$ be an acute non-isosceles triangle with circumcircle $\omega$, circumcenter $O$ and orthocenter $H$. We draw a line perpendicular to $AH$ through $O$ and a line perpendicular to $AO$ through $H$. Prove that the points of intersection of these lines with sides $AB$ and $AC$ lie on a circle, which is tangent to $\omega$. Proposed by A. Kuznetsov

In a country there are $n>100$ cities and initially no roads. The government randomly determined the cost of building a two-way road between any two cities, using all amounts from $1$ to $\frac{n(n-1)}{2}$ thalers once (all options are equally likely). The mayor of each city chooses the cheapest of the $n-1$ roads emanating from that city and it is built (this may be the mutual desired of the mayors of both cities being connected, or only one of the two). After the construction of these roads, the cities are divided into $M$ connected components (between cities of the same connected component, you can get along the constructed roads, possibly via other cities, but this is not possible for cities of different components). Find the expected value of the random variable $M$. Proposed by F. Petrov

Prove that there exists $c>0$ such that for any odd prime $p=2k+1$, the numbers $1^0, 2^1,3^2,\dots,k^{k-1}$ give at least $c\sqrt{p}$ distinct residues modulo $p$. Proposed by M. Turevsky, I. Bogdanov