We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{} m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\] [i]V. T. Sos[/i]

In an excavation in ancient Rome an unusual clock with $18$ divisions marked with Roman numerals (see figure). Unfortunately the watch was broken into $5$ pieces. The sum of the numbers on each piece was the same. Show how he could be broken the clock. [img]https://cdn.artofproblemsolving.com/attachments/7/a/6e83df1bb7adb13305239a152ac95a4a96f556.png[/img]

Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half?

Robot "Mag-o-matic" manipulates $101$ glasses, displaced in a row whose positions are numbered from $1$ to $101$. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form $(a;b,c)$, which it interprets as "consider the glass in position $a$: if it contains a ball, then switch the glasses in positions $b$ and $c$ (together with their own content), otherwise move on to the following instruction" (it means that $a,\,b,\,c$ are integers between $1$ and $101$, with $b$ and $c$ different from each other but not necessarily different from $a$). A $\emph{programme}$ is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one. A subset $S\subseteq \{0,\,1,\,2,\dots,\,101\}$ is said to be $\emph{identifiable}$ if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position $1$ contains a ball if and only if the number of glasses containing a ball is an element of $S$. (a) Prove that the subset $S$ of the odd numbers is identifiable. (b) Determine all subsets $S$ that are identifiable.

In a rectangle with vertices $(0, 0), (0, 2), (n,0),(n, 2)$, ($n$ is a positive integer) find the number of longest paths starting from $(0, 0)$ and arriving at $(n, 2)$ which satisfy the following: $\bullet$ At each movement, you can move right, up, left, down by $1$. $\bullet$ You cannot visit a point you visited before. $\bullet$ You cannot move outside the rectangle.

In each $ 2007^{2}$ unit squares on chess board whose size is $ 2007\times 2007$, there lies one coin each square such that their "heads" face upward. Consider the process that flips four consecutive coins on the same row, or flips four consecutive coins on the same column. Doing this process finite times, we want to make the "tails" of all of coins face upward, except one that lies in the $ i$th row and $ j$th column. Show that this is possible if and only if both of $ i$ and $ j$ are divisible by $ 4$.

Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.

We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$, a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$, such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.

In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be[i] k -directionally[/i] connected to the city $B$, if : we can orient at most $k$ roads such that after[i] arbitrary[/i] orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be[i] k - directionally[/i] connected. What is the minimal value of $k$?

Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set $\{0,1,\ldots,10\}.$ Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?

The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?

Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.

In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that (i) every student goes to the same destination as their best friend; (ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to $1$.

Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$ Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$ [i]Edited by orl.[/i]

How many integers $n$ from $1$ to $2020$, inclusive, are there such that $2020$ divides $n^2 + 1$?

Each real number greater than $1$ is coloured red or blue with both colours being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+b$ and $ab$ are of different colours.

A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square. Such a board is called [i] interesting[/i] if the following conditions hold: $\circ$ In all unit squares below the main diagonal, the number $0$ is written; $\circ$ Positive integers are written in all other unit squares. $\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers are actually the numbers $1,2,...,2n$ (not necessarily in that order). $a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board. $b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.

A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one. Find the maximum possible number of distinct prime divisors of $n$

Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.

Given a set $L$ of lines in general position in the plane (no two lines in $L$ are parallel, and no three lines are concurrent) and another line $\ell$, show that the total number of edges of all faces in the corresponding arrangement, intersected by $\ell$, is at most $6|L|$. [i]Chazelle et al., Edelsbrunner et al.[/i]

In some squares of a $10 \times 10$ board, a piece is placed in such a way that the following property is satisfied: For each square that has a piece, the number of pieces placed in the same row must be greater than or equal to the number of pieces placed in the same column. How many tiles can there be on the board? Give all chances.

Let $n$ and $k$ be positive integers. Given $n$ closed discs in the plane such that no matter how we choose $k + 1$ of them, there are always two of the chosen discs that have no common point. Prove that the $n$ discs can be partitioned into at most $10k$ classes such that any two discs in the same class have no common point.

Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that: (i) the first cell in the sequence is a valley, (ii) each subsequent cell in the sequence is adjacent to the previous cell, and (iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square. Author: Nikola Petrović