Let $S=\{1,2,\cdots ,n\}$ where $n$ is an odd integer. Let $f$ be a function defined on $\{(i,j): i\in S, j \in S\}$ taking values in $S$ such that (i) $f(s,r)=f(r,s)$ for all $r,s \in S$ (ii) $\{f(r,s): s\in S\}=S$ for all $r\in S$ Show that $\{f(r,r): r\in S\}=S$

Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$

Let $a_1,a_2,..., a_n$ be real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $|a_1| + |a_2 | + ... + |a_n | = 1$. Prove that $$ |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2} $$

Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest . Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$. In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese[/url].

Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, \[f(xf(y))+f(y)=f(x+y)+f(xy).\] [i]Milan Haiman[/i]

Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.

Let $(x_n)_{n\in \mathbb{N}}$ be a sequence defined recursively such that $x_1=\sqrt{2}$ and $$x_{n+1}=x_n+\frac{1}{x_n}\text{ for }n\in \mathbb{N}.$$ Prove that the following inequality holds: $$\frac{x_1^2}{2x_1x_2-1}+\frac{x_2^2}{2x_2x_3-1}+\dotsc +\frac{x_{2018}^2}{2x_{2018}x_{2019}-1}+\frac{x_{2019}^2}{2x_{2019}x_{2020}-1}>\frac{2019^2}{x_{2019}^2+\frac{1}{x_{2019}^2}}.$$ [i]Proposed by Ivan Novak[/i]

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

A function $g$ defined for all positive integers $n$ satisfies [list][*]$g(1) = 1$; [*]for all $n\ge 1$, either $g(n+1)=g(n)+1$ or $g(n+1)=g(n)-1$; [*]for all $n\ge 1$, $g(3n) = g(n)$; and [*]$g(k)=2001$ for some positive integer $k$.[/list] Find, with proof, the smallest possible value of $k$.

Determine all integers \( x \) for which the expression \( x^2 + 10x + 160 \) is a perfect square.

Compute the sum of all positive integers $n$ for which the expression \[\frac{n+7}{\sqrt{n-1}}\] is an integer.

Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that \[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\] Prove that $(a_n)_{n\ge0}$ is a geometric sequence. [i]Lucian Dragomir[/i]

[b]p1.[/b] In the correct identity $(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)$ two numbers were replaced with dots. What were these numbers? [b]p2.[/b] A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\%$ of the amount currently available, on the other - $20\%$, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? [img]https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png[/img] [b]p3.[/b] Find the angle between the hour and minute hands at $7$ hours $38$ minutes. [b]p4.[/b] The lottery game is played as follows. A random number from $1$ to $1000$ is selected. If it is divisible by $2$, they pay a ruble, if it is divisible by $10$ - two rubles, by $12$ - four rubles, by $20$ - eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options. [b]p5.[/b]The sum of the digits of a positive integer $x$ is equal to $n$. Prove that between $x$ and $10x$ you can find an integer whose sum of digits is $ n + 5$. [b]p6.[/b] $9$ people took part in the campaign, which lasted $12$ days. There were $3$ people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all $12$ days they were able to appoint three people on duty, taking into account this requirement. Could this be so? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

[b]p1.[/b] Consider a $4 \times 4$ lattice on the coordinate plane. At $(0,0)$ is Mori’s house, and at $(4,4)$ is Mori’s workplace. Every morning, Mori goes to work by choosing a path going up and right along the roads on the lattice. Recently, the intersection at $(2, 2)$ was closed. How many ways are there now for Mori to go to work? [b]p2.[/b] Given two integers, define an operation $*$ such that if a and b are integers, then a $*$ b is an integer. The operation $*$ has the following properties: 1. $a * a$ = 0 for all integers $a$. 2. $(ka + b) * a = b * a$ for integers $a, b, k$. 3. $0 \le b * a < a$. 4. If $0 \le b < a$, then $b * a = b$. Find $2017 * 16$. [b]p3.[/b] Let $ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $CA = 15$. Let $A'$, $B'$, $C'$, be the midpoints of $BC$, $CA$, and $AB$, respectively. What is the ratio of the area of triangle $ABC$ to the area of triangle $A'B'C'$? [b]p4.[/b] In a strange world, each orange has a label, a number from $0$ to $10$ inclusive, and there are an infinite number of oranges of each label. Oranges with the same label are considered indistinguishable. Sally has 3 boxes, and randomly puts oranges in her boxes such that (a) If she puts an orange labelled a in a box (where a is any number from 0 to 10), she cannot put any other oranges labelled a in that box. (b) If any two boxes contain an orange that have the same labelling, the third box must also contain an orange with that labelling. (c) The three boxes collectively contain all types of oranges (oranges of any label). The number of possible ways Sally can put oranges in her $3$ boxes is $N$, which can be written as the product of primes: $$p_1^{e_1} p_2^{e_2}... p_k^{e_k}$$ where $p_1 \ne p_2 \ne p_3 ... \ne p_k$ and $p_i$ are all primes and $e_i$ are all positive integers. What is the sum $e_1 + e_2 + e_3 +...+ e_k$? [b]p5.[/b] Suppose I want to stack $2017$ identical boxes. After placing the first box, every subsequent box must either be placed on top of another one or begin a new stack to the right of the rightmost pile. How many different ways can I stack the boxes, if the order I stack them doesn’t matter? Express your answer as $$p_1^{e_1} p_2^{e_2}... p_n^{e_n}$$ where $p_1, p_2, p_3, ... , p_n$ are distinct primes and $e_i$ are all positive integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many positive roots does polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have such that $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 1001 \qquad\textbf{e)}\ 2002 $

The system $\begin{cases} x^2 - y^2 = 0 \\ (x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?

Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$ \[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

[b]Problem 3[/b] : Let $x_1,x_2,\cdots,x_{2022}$ be non-negative real numbers such that $$x_k + x_{k+1}+x_{k+2} \leq 2$$ for all $k = 1,2,\cdots,2020$. Prove that $$\sum_{k=1}^{2020}x_kx_{k+2}\leq 1010$$

Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$. (D. Bazylev)