We have a line and $1390$ points around it such that the distance of each point to the line is less than $1$ centimeters and the distance between any two points is more than $2$ centimeters. prove that there are two points such that their distance is at least $10$ meters ($1000$ centimeters).

p1. It is known that $m$ and $n$ are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number $4$ and the number $5$. The number $59$ is a prime factor of $m$. The remainder of the division of $n$ by $38$ is $ 1$. If the difference between $m$ and $n$ is not more than $2015$. determine all possible pairs of numbers $(m,n)$. p2. It is known that the equation $ax^2 + bx + c = $0 with $a> 0$ has two different real roots and the equation $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$ has no real roots. Is it true that $ad^2 + 2ad^2 <4bc + 16c^3$ ? p3. A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements. p4. It is known that two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at point $P$ and at point $Q$. The point $U$ lies on the circle $L_2$ so that the line segment $PU$ is a diameter of the circle $L_2$. The point $T$ lies at the extension of the line segment $PQ$ such that the area of ​​the quadrilateral $QTUN$ is $792/25$ units of area. Determine the length of the $QT$. p5. An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e. $V_n - V_{n+1} = a L_n$, for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of ​​the property, that is $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$ , where $k$ is a positive constant. If $V_1=\frac{27}{64} V_0$ and the ice ball melts totally at exactly $h$ seconds, determine the value of $h$.

Show that the product of every $k$ consecutive members of the Fibonacci sequence is divisible by $f_1f_2\ldots f_k$ (where $f_0=0$ and $f_1=1$).

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

The faces of an icosahedron are painted into $5$ colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.

[u]Round 5[/u] [b]5.1[/b] A circle with a radius of $1$ is inscribed in a regular hexagon. This hexagon is inscribed in a larger circle. If the area that is outside the hexagon but inside the larger circle can be expressed as $\frac{a\pi}{b} - c\sqrt{d}$, where $a, b, c, d$ are positive integers, $a, b$ are relatively prime, and no prime perfect square divides into $d$. find the value of $a + b + c + d$. [b]5.2[/b] At a dinner party, $10$ people are to be seated at a round table. If person A cannot be seated next to person $B$ and person $C$ must be next to person $D$, how many ways can the 10 people be seated? Consider rotations of a configuration identical. [b]5.3[/b] Let $N$ be the sum of all the positive integers that are less than $2022$ and relatively prime to $1011$. Find $\frac{N}{2022}$. [u]Round 6[/u] [b]6.1[/b] The line $y = m(x - 6)$ passes through the point $ A$ $(6, 0)$, and the line $y = 8 -\frac{x}{m}$ pass through point $B$ $(0,8)$. The two lines intersect at point $C$. What is the largest possible area of triangle $ABC$? [b]6.2[/b] Let $N$ be the number of ways there are to arrange the letters of the word MATHEMATICAL such that no two As can be adjacent. Find the last $3$ digits of $\frac{N}{100}$. [b]6.3[/b] Find the number of ordered triples of integers $(a, b, c)$ such that $|a|, |b|, |c| \le 100$ and $3abc = a^3 + b^3 + c^3$. [u]Round 7[/u] [b]7.1[/b] In a given plane, let $A, B$ be points such that $AB = 6$. Let $S$ be the set of points such that for any point $C$ in $S$, the circumradius of $\vartriangle ABC$ is at most $6$. Find $a + b + c$ if the area of $S$ can be expressed as $a\pi + b\sqrt{c}$ where $a, b, c$ are positive integers, and $c$ is not divisible by the square of any prime. [b]7.2[/b] Compute $\sum_{1\le a<b<c\le 7} abc$. [b]7.3[/b] Three identical circles are centered at points $A, B$, and $C$ respectively and are drawn inside a unit circle. The circles are internally tangent to the unit circle and externally tangent to each other. A circle centered at point $D$ is externally tangent to circles $A, B$, and $C$. If a circle centered at point $E$ is externally tangent to circles $A, B$, and $D$, what is the radius of circle $E$? The radius of circle $E$ can be expressed as $\frac{a\sqrt{b}-c}{d}$ where $a, b, c$, and d are all positive integers, gcd(a, c, d) = 1, and b is not divisible by the square of any prime. What is the sum of $a + b + c + d$? [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of unused Algebra problems in our problem bank. Let $B$ be the number of times the letter ’b’ appears in our problem bank. Let M be the median speed round score. Finally, let $C$ be the number of correct answers to Speed Round $1$. Estimate $$A \cdot B + M \cdot C.$$ Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2826128p24988676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the integral lattice $\mathbb{Z}^n$, $n \geq 2$, in the Euclidean $n$-space. Define a [i]line[/i] in $\mathbb{Z}^n$ to be a set of the form $a_1 \times \cdots \times a_{k-1} \times \mathbb{Z} \times a_{k+1} \times \cdots \times a_n$ where $k$ is an integer in the range $1,2,\ldots,n$, and the $a_i$ are arbitrary integers. A subset $A$ of $\mathbb{Z}^n$ is called [i]admissible[/i] if it is non-empty, finite, and every [i]line[/i] in $\mathbb{Z}^{n}$ which intersects $A$ contains at least two points from $A$. A subset $N$ of $\mathbb{Z}^n$ is called [i]null[/i] if it is non-empty, and every [i]line[/i] in $\mathbb{Z}^n$ intersects $N$ in an even number of points (possibly zero). [b](a)[/b] Prove that every [i]admissible[/i] set in $\mathbb{Z}^2$ contains a [i]null[/i] set. [b](b)[/b] Exhibit an [i]admissible[/i] set in $\mathbb{Z}^3$ no subset of which is a [i]null[/i] set .

In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$

Prove that no convex $13$-gon can be cut into parallelograms.

From numbers $ 1,2,3,...,37$ we randomly choose 10 numbers. Prove that among these exist four distinct numbers, such that sum of two of them equals to the sum of other two.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

The transportation ministry has just decided to pay $80$ companies to repair $2400$ roads. These roads connects $100$ cities. Each road is between two cities and there is at most one road between any two cities. Each company must repair exactly $30$ roads, and each road is repaired by exactly one company. For a company to repair a road, it is necessary for the company to set up stations at the both cities on its endpoints. Prove that there are at least $8$ companies stationed at one city.

$1399$ points and some chords between them is given. $a)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase [u]exactly one[/u] of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions. $b)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase [u]both[/u] of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions. [i]Proposed by Afrouz Jabalameli, Abolfazl Asadi[/i]

There are $42$ students taking part in the Team Selection Test. It is known that every student knows exactly $20$ other students. Show that we can divide the students into $2$ groups or $21$ groups such that the number of students in each group is equal and every two students in the same group know each other.

There are $33!$ empty boxes labeled from $1$ to $33!$. In every move, we find the empty box with the smallest label, then we transfer $1$ ball from every box with a smaller label and we add an additional $1$ ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after $33!$ moves.

Annie has a permutation $(a_1, a_2, \dots ,a_{2019})$ of $S=\{1,2,\dots,2019\}$, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an $n$-tuple $(y_1, y_2, \dots, y_{2019})$ of integers in $S$, and then Annie gives the number of indices $i\in S$ such that $a_i=y_i$. (a) Show that Yannick can always guess Annie's permutation with at most $1200000$ guesses. (b) Show that Yannick can always guess Annie's permutation with at most $24000$ guesses. [i]Yannick Yao[/i]

a) Can you pose the numbers $0,1,...,9$ on the circumference in such a way, that the difference between every two neighbours would be either $3$ or $4$ or $5$? b) The same question, but about the numbers $0,1,...,13$.

With inspiration drawn from the rectilinear network of streets in [i]New York[/i] , the [i]Manhattan distance[/i] between two points $(a,b)$ and $(c,d)$ in the plane is defined to be \[|a-c|+|b-d|\] Suppose only two distinct [i]Manhattan distance[/i] occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?

The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values. [i]Proposed by A. Golovanov[/i]

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$

In a "Hanger Party", the guests are initially dressed. In certain moments, the host chooses a guest, and the chosen guest and all his friends will wear its respective clothes if they are naked, and undress it if they are dressed. It is possible that, in some moment, the guests are naked, independent of their mutual friendships? (Suppose friendship is reciprocal.)

In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

21 players participated in a tennis tournament, in which each pair of players played exactly once and each game had a winner (no ties are allowed). The organizers of the tournament found out that each player won at least 9 games, and lost at least 9. In addition, they discovered cases of three players $A,B,C$ in which $A$ won against $B$, $B$ won against $C$ and $C$ won against $A$, and called such triples "problematic". [b]a)[/b] What is the maximum possible number of problematic triples? [b]b)[/b] What is the minimum possible number of problematic triples?

We are given n objects of identical appearance, but different mass, and a balance which can be used to compare any two objects (but only one object can be placed in each pan at a time). How many times must we use the balance to find the heaviest object and the lightest object?