Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?

Find all the odd naturals whose indicator is the same as $1990$. We clarify that, if a natural decomposes into prime factors in the form $\Pi_{j=1}^r p_j^{a_j}$, define the [i]indicator [/i] as : $\phi (n) = r\Pi_{j=1}^r p_j^{a_j-1} (p_j + 1)$. [hide=official wording for first sentence]Encuentre todos los naturales impares cuyo indicador es el mismo que el de 1990.[/hide]

Given a natural number $n\ge 3$, determine all strictly increasing sequences $a_1<a_2<\cdots<a_n$ such that $\text{gcd}(a_1,a_2)=1$ and for any pair of natural numbers $(k,m)$ satisfy $n\ge m\ge 3$, $m\ge k$, $$\frac{a_1+a_2+\cdots +a_m}{a_k}$$ is a positive integer.

Let $x,y,z\in \mathbb{R}^*$ such that $xy,yz,zx\in \mathbb{Q}$. a) Prove that $x^2+y^2+z^2$ is rational; b) If $x^3+y^3+z^3$ is rational, prove that $x,y,z$ are rational. [i]Marius Ghergu[/i]

On a table, we have ten thousand matches, two of which are inside a bowl. Anna and Bernd play the following game: They alternate taking turns and Anna begins. A turn consists of counting the matches in the bowl, choosing a proper divisor $d$ of this number and adding $d$ matches to the bowl. The game ends when more than $2024$ matches are in the bowl. The person who played the last turn wins. Prove that Anna can win independently of how Bernd plays. [i](Richard Henner)[/i]

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

The sequence $(L_n)$ is given by $L_0=2$, $L_1=1$, and $L_{n+1}=L_n+L_{n-1}$ for $n\ge1$. Prove that if a prime number $p$ divides $L_{2k}-2$ for $k\in\mathbb N$, then $p$ also divides $L_{2k+1}-1$.

Let $p_n$ be the $n$-th prime. ($p_1=2$) Define the sequence $(f_j)$ as follows: - $f_1=1, f_2=2$ - $\forall j\ge 2$: if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$ - $\forall j\ge 2$: if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$ (a) Show that all $f_i$ are different (b) from which index onwards are all $f_i$ at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?

Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

Let $n,d$ be natural numbers. Prove that there is an arithmetic sequence of positive integers. $$a_1,a_2,...,a_n$$ with common difference of $d$ and $a_i$ with prime factor greater than or equal to $i$ for all values $i=1,2,...,n$. [i](nooonuii)[/i]

Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$. [i](41th Austrian Mathematical Olympiad, regional competition, problem 4)[/i]

Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

Estimate the number of $1$s in the hexadecimal representation of $2020!$. If $E$ is your estimate and $A$ is the correct answer, you will receive $\max (25 - 0.5|A - E|, 0)$ points, rounded to the nearest integer.

Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and \[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]

Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]

For an integer $ n \geq 2 $, let $ s (n) $ be the sum of positive integers not exceeding $ n $ and not relatively prime to $ n $. a) Prove that $ s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) $, where $ \varphi (n) $ is the number of integers positive cannot exceed $ n $ and are relatively prime to $ n $. b) Prove that there is no integer $ n \geq 2 $ such that $ s (n) = s (n + 2021) $

For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$

Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. [quote]For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1[/quote] The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ . Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$: $P(x^2-y^2) = P(x+y)P(x-y)$.

Let $f(n)$ count the number of values $0\le k\le n^2$ such that $43\nmid\binom{n^2}{k}$. Find the least positive value of $n$ such that $$43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)$$ [i]Proposed by Adam Bertelli[/i]