Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

$32$ stones, with pairwise different weights, and lever scales without weights are given. How to determine by $35$ scaling, which stone is the heaviest and which is the second by weight?

In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo). What is the maximal possible number of edges in such a graph?

Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$.

Show that in a tournament of $799$ teams (every team plays with every other team for a win or loss), there exist $14$ teams such that the first seven teams have each defeated the remaining teams.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

A regular octagon $P$ is given whose incircle $k$ has diameter $1$. About $k$ is circumscribed a regular $16$-gon, which is also inscribed in $P$, cutting from $P$ eight isosceles triangles. To the octagon $P$, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$-gon so obtained is said to be $P'$. Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$, one of the $11$-gons $P'$ contains all of $M$.

A football is covered by some polygonal pieces of leather which are sewed up by three different colors threads. It features as follows: i) any edge of a polygonal piece of leather is sewed up with an equal-length edge of another polygonal piece of leather by a certain color thread; ii) each node on the ball is vertex to exactly three polygons, and the three threads joint at the node are of different colors. Show that we can assign to each node on the ball a complex number (not equal to $1$), such that the product of the numbers assigned to the vertices of any polygonal face is equal to $1$.

Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].\] Let those two collections satisfy the following properties: (i) On each row of $P$, from left to right, the numbers are non-increasing, (ii) On each column of $Q$, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in $P$ equals to the same row in $Q$, (iv) The sum of numbers on the column in $P$ equals to the same column in $Q$. Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$. [i]Proposer: Stefanus Lie[/i]

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)

Suppose you started at the origin on the number line in a coin-flipping game. Every time you flip a heads, you move forward one step, otherwise you move back one step. However, there are walls at positions $8$ and $-8$; if you are at these positions and your coin flip dictates that you should move past them, instead you must stay. What is the expected number of coin flips needed to have visited both walls?

$22$ football players took part in the football training. They were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]

Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)

Let us call "[i]big[/i]" a triangle with all sides longer than $1$. Given a equilateral triangle with all the sides equal to $5$. Prove that: a) You can cut $100$ [i]big [/i] triangles out of given one. b) You can divide the given triangle onto $100$ [i]big [/i] nonintersecting ones fully covering the initial one. c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex. d) The same as c), but the initial triangle has the side $3$.

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

In each cell of a $4 \times 4$ table a digit $1$ or $2$ is written. Suppose that the sum of the digits in each of the four $3 \times 3$ sub-tables is divisible by $4$, but the sum of the digits in the entire table is not divisible by $4$. Find the greatest and the smallest possible value of the sum of the $16$ digits.

Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold: 1. each color is used at least once; 2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational. Find the least possible value of $N$. [i]~Sutanay Bhattacharya[/i]

[u]Round 5[/u] [b]p13.[/b] Express the number $3024_8$ in base $2$. [b]p14.[/b] $\vartriangle ABC$ has a perimeter of $10$ and has $AB = 3$ and $\angle C$ has a measure of $60^o$. What is the maximum area of the triangle? [b]p15.[/b] A weighted coin comes up as heads $30\%$ of the time and tails $70\%$ of the time. If I flip the coin $25$ times, howmany tails am I expected to flip? [u]Round 6[/u] [b]p16.[/b] A rectangular box with side lengths $7$, $11$, and $13$ is lined with reflective mirrors, and has edges aligned with the coordinate axes. A laser is shot from a corner of the box in the direction of the line $x = y = z$. Find the distance traveled by the laser before hitting a corner of the box. [b]p17.[/b] The largest solution to $x^2 + \frac{49}{x^2}= 2018$ can be represented in the form $\sqrt{a}+\sqrt{b}$. Compute $a +b$. [b]p18.[/b] What is the expected number of black cards between the two jokers of a $54$ card deck? [u]Round 7[/u] p19. Compute ${6 \choose 0} \cdot 2^0 + {6 \choose 1} \cdot 2^1+ {6 \choose 2} \cdot 2^2+ ...+ {6 \choose 6} \cdot 2^6$. [b]p20.[/b] Define a sequence by $a_1 =5$, $a_{n+1} = a_n + 4 * n -1$ for $n\ge 1$. What is the value of $a_{1000}$? [b]p21.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B = 15^o$ and $\angle C = 30^o$. Let $D$ be the point such that $\vartriangle ADC$ is an isosceles right triangle where $D$ is in the opposite side from $A$ respect to $BC$ and $\angle DAC = 90^o$. Find the $\angle ADB$. [u]Round 8[/u] [b]p22.[/b] Say the answer to problem $24$ is $z$. Compute $gcd (z,7z +24).$ [b]p23.[/b] Say the answer to problem $22$ is $x$. If $x$ is $1$, write down $1$ for this question. Otherwise, compute $$\sum^{\infty}_{k=1} \frac{1}{x^k}$$ [b]p24.[/b] Say the answer to problem $23$ is $y$. Compute $$\left \lfloor \frac{y^2 +1}{y} \right \rfloor$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.

A bookshelf contains $n$ volumes, labelled $1$ to $n$, in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label $k$, takes it out, and inserts it in the $k$-th position. For example, if the bookshelf contains the volumes $1,3,2,4$ in that order, the librarian could take out volume $2$ and place it in the second position. The books will then be in the correct order $1,2,3,4$. (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order. (b) What is the largest number of steps that this process can take?

Tracy has been baking a rectangular cake whose surface is dissected by grid lines in square fields. The number of rows is $ 2^n$ and the number of columns is $ 2^{n \plus{} 1}$ where $ n \geq 1, n \in \mathbb{N}.$ Now she covers the fields with strawberries such that each row has at least $ 2n \plus{} 2$ of them. Show that there four pairwise distinct strawberries $ A,B,C$ and $ D$ which satisfy those three conditions: (a) Strawberries $ A$ and $ B$ lie in the same row and $ A$ further left than $ B.$ Similarly $ D$ lies in the same row as $ C$ but further left. (b) Strawberries $ B$ and $ C$ lie in the same column. (c) Strawberries $ A$ lies further up and further left than $ D.$

On each side of an equilateral triangle of side $n \ge 1$ consider $n - 1$ points that divide the sides into $n$ equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length $1$. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. Colombia 1997