Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

The four circles in the figure determine 10 bounded regions. $10$ numbers summing to $100$ are written in these regions, one in each region. The sum of the numbers contained in each circle is equal to $S$ (the same quantity for each of the four circles). Determine the greatest and smallest possible values of $S$.

find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equations: $x+y+z=7; x^2+y^2+z^2=27; xyz=5$.

Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.

A function $f$ defined on integers such that $f (n) =n + 3$ if $n$ is odd $f (n) = \frac{n}{2}$ if $n$ is even If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$.

$a_1\geq a_2\geq... \geq a_{100n}>0$ If we take from $(a_1,a_2,...,a_{100n})$ some $2n+1$ numbers $b_1\geq b_2 \geq ... \geq b_{2n+1}$ then $b_1+...+b_n > b_{n+1}+...b_{2n+1}$ Prove, that $$(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}$$

Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality: $$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given three nonzero real numbers $a,b,c,$ such that $a>b>c$, prove the equation has at least one real root. $$\frac{1}{x+a}+\frac{1}{x+b}+\frac{1}{x+c}-\frac{3}{x}=0$$ @below sorry, I believe I fixed it with the added constraint.

Given several nonnegative integers (repetitions allowed), the allowed operation is to choose a positive integer $a$ and replace each number $b$ greater than or equal to $a$ by $b-a$ (the numbers $a$ , if any, are replaced by $0$). Initially, the integers from $1$ are written on the blackboard until $2013$ inclusive. After a few operations the numbers on the board have a sum equal to $10$. Determine what the numbers that remained on the board could be. Find all the possibilities.

For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if $\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and $\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$. Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$. [i]Proposed by Noga Alon and Jean Bourgain[/i]

Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations $$\frac{x}{\sqrt{x^2 + y^2}}-\frac{1}{x}= 7 \,\,\, \text{and} \,\,\, \frac{y}{\sqrt{x^2 + y^2}}+\frac{1}{y}=4 $$

Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows: $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a. Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b. Find all values of $a$ in the equality case.

Find all functions $f$ such that \[f(f(x) + y) = f(x^2-y) + 4yf(x)\] for all real numbers $x$ and $y$.

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

Determine the type of triangle if the lengths of its sides $a, b, c$ satisfy the relation $$a^4 + b^4 + c^4 = a^2b^2 + b^2c^2 + c^2a^2$$

Let $x$ and $y$ are real numbers such that $$\begin{cases} x + y = 2 \\ x^3 + y^3 = 3\end{cases} $$ What is $x^2+y^2$?

Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.

Let $P$ be a polynomial of degree $n$ satisfying \[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\] Determine $P(n + 1).$

Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions $f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s). Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$

Let $n \ge 2$ be a fixed integer. [list=a] [*]Determine the largest positive integer $m$ (in terms of $n$) such that there exist complex numbers $r_1$, $\dots$, $r_n$, not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of $m$, find all possible values of \[ \prod\limits_{k=1}^n (r_k^{m+1}+1). \] [/list] [i]Kaixin Wang[/i]

A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.

If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$

Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.

The sequence $(a_n)$ is defined by $a_1=1$ and $a_n=a_{n-1}+\frac{1}{n^3}$ for $n>1.$ (a) Prove that $a_n<\frac{5}{4}$ for all $n.$ (b) Given $\epsilon>0$, find the smallest natural number $n_0$ such that ${\mid a_{n+1}-a_n}\mid<\epsilon$ for all $n>n_0.$