It is given the function $f:\left( \mathbb{R} - \{0\} \right) \times \left( \mathbb{R}-\{0\} \right) \to \mathbb{R}$ such that $f(a,b)= \left| \frac{|b-a|}{|ab|}+\frac{b+a}{ab}-1 \right|+ \frac{|b-a|}{|ab|}+ \frac{b+a}{ab}+1$ where $a,b \not=0$. Prove that: \[ f(a,b)=4 \cdot \text{max} \left\{\frac{1}{a},\frac{1}{b},\frac{1}{2} \right\}\]

A finite set $M$ of real numbers is called [i]special[/i] if $M$ has at least two elements and the following condition is true: If $a$ and $b$ are distinct elements of $M$ then $5\sqrt{|a|}-\frac{2b}{3}$ is also a element of $M$. a) Determine if there is a special set with (exactly) two elements. b) Determine if there is a special set with three (or more) elements such that all elements are positive.

Let $\frac{1}{1-x-x^2-x^3} =\sum^{\infty}_{i=0} a_nx^n$, for what positive integers $n$ does $a_{n-1} = n^2$?

Find the values of $ x $ that satisfy the inequality $$ \sqrt{x} - \sqrt{x- a} > 2,$$ where $ a $ is a gicen poistive number.

For positive real numbers $a_1, a_2, ..., a_n$ with product 1 prove: $$\left(\frac{a_1}{a_2}\right)^{n-1}+\left(\frac{a_2}{a_3}\right)^{n-1}+...+\left(\frac{a_{n-1}}{a_n}\right)^{n-1}+\left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^{2}+a_2^{2}+...+a_n^{2}$$ Proposed by Andrei Eckstein

[b]a)[/b] Show that two real numbers $ x,y>1 $ chosen so that $ x^y=y^x, $ are equal or there exists a positive real number $ m\neq 1 $ such that $ x=m^{\frac{1}{m-1}} $ and $ y=m^{\frac{m}{m-1}} . $ [b]b)[/b] Solve in $ \left( 1,\infty \right)^2 $ the equation: $ x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} . $

The World Cup, featuring $17$ teams from Europe and South America, as well as $15$ other teams that honestly don’t have a chance, is a soccer tournament that is held once every four years. As we speak, Croatia andMorocco are locked in a battle that has no significance whatsoever on the winner, but if you would like live score updates nonetheless, feel free to ask your proctor, who has no obligation whatsoever to provide them. [b]p1.[/b] During the group stage of theWorld Cup, groups of $4$ teams are formed. Every pair of teams in a group play each other once. Each team earns $3$ points for each win and $1$ point for each tie. Find the greatest possible sum of the points of each team in a group. [b]p2.[/b] In the semi-finals of theWorld Cup, the ref is bad and lets $11^2 = 121$ players per team go on the field at once. For a given team, one player is a goalie, and every other player is either a defender, midfielder, or forward. There is at least one player in each position. The product of the number of defenders, midfielders, and forwards is a mulitple of $121$. Find the number of ordered triples (number of defenders, number of midfielders, number of forwards) that satisfy these conditions. [b]p3.[/b] Messi is playing in a game during the Round of $16$. On rectangular soccer field $ABCD$ with $AB = 11$, $BC = 8$, points $E$ and $F$ are on segment $BC$ such that $BE = 3$, $EF = 2$, and $FC = 3$. If the distance betweenMessi and segment $EF$ is less than $6$, he can score a goal. The area of the region on the field whereMessi can score a goal is $a\pi +\sqrt{b} +c$, where $a$, $b$, and $c$ are integers. Find $10000a +100b +c$. [b]p4.[/b] The workers are building theWorld Cup stadium for the $2022$ World Cup in Qatar. It would take 1 worker working alone $4212$ days to build the stadium. Before construction started, there were 256 workers. However, each day after construction, $7$ workers disappear. Find the number of days it will take to finish building the stadium. [b]p5.[/b] In the penalty kick shootout, $2$ teams each get $5$ attempts to score. The teams alternate shots and the team that scores a greater number of times wins. At any point, if it’s impossible for one team to win, even before both teams have taken all $5$ shots, the shootout ends and nomore shots are taken. If each team does take all $5$ shots and afterwards the score is tied, the shootout enters sudden death, where teams alternate taking shots until one team has a higher score while both teams have taken the same number of shots. If each shot has a $\frac12$ chance of scoring, the expected number of times that any team scores can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let {$a_1, a_2,\cdots,a_n$} be a set of $n$ real numbers whos sym equals S. It is known that each number in the set is less than $\frac{S}{n-1}$. Prove that for any three numbers $a_i$, $a_j$ and $a_k$ in the set, $a_i+a_j>a_k$.

The AIME Triathlon consists of a half-mile swim, a $30$-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$ [i]Proposed by N. Safaei (Iran)[/i]

a) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}$ b) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}$ How does the solution vary for distinct values of $n$?

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Let $b_1 = 1$ and $ b_{n+1} = 1 + \frac{1}{n(n+1)b_1b_2...b_n}$ for $n \ge 1$. Find $b_12$.

Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$. Alexandru Gırban

Show that for each non-negative integer $n$ there are unique non-negative integers $x$ and $y$ such that we have \[n=\frac{(x+y)^2+3x+y}{2}.\]

The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence: $$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$ Given that $a_{2021} = 2021$, find $a_1$.

Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?

The geometric mean of a set of $m$ non-negative numbers is the $m$-th root of the product of these numbers. For which positive values of ​​$n$, is there a finite set $S_n$ of $n$ positive integers different such that the geometric mean of any subset of $S_n$ is an integer?

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.

Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: $\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad \textbf{(B)}\ f(n)\qquad \textbf{(C)}\ 2f(n)+1 \qquad \textbf{(D)}\ f^2(n) \qquad \textbf{(E)}\ \dfrac{1}{2}(f^2(n)-1)$

Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]

Suppose $z=a+bi$ is a solution of the polynomial equation $c_4z^4+ic_3z^3+c_2z^2+ic_1z+c_0=0$, where $c_0$, $c_1$, $c_2$, $c_3$, $a$, and $b$ are real constants and $i^2=-1$. Which of the following must also be a solution? $\textbf{(A) } -a-bi\qquad \textbf{(B) } a-bi\qquad \textbf{(C) } -a+bi\qquad \textbf{(D) }b+ai \qquad \textbf{(E) } \text{none of these}$