Find all real numbers $a,b,c,d,e$ in the interval $[-2,2]$, that satisfy: \begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}

Find all natural numbers $n$ for which there is a permutation $\sigma$ of $\{1,2,\ldots, n\}$ that satisfies: \[ \sum_{i=1}^n \sigma(i)(-2)^{i-1}=0 \]

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

Determine all polynomials $P{}$ with integer coefficients, satisfying $0 \leqslant P (n) \leqslant n!$ for all non-negative integers $n$. [i]Andrei Chirita[/i]

Find all reals $x$ satisfying $0 \le x \le 5$ and $\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.

If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?

If we have a set $M$ of $2019$ real numbers such that for every even $a$, $b$ of numbers of $M$ it is verified that $a^2+b \sqrt2$ is a rational number. Show that for all $a$ of $M$, $a\sqrt2$ is a rational number.

Let $\phi =\frac{1}{2019}$. Define $$g_n =\begin{cases} 0 & \text{ if} \,\,\,\, round (n\phi) = round \,\,\,\, ((n - 1)\phi) \\ 1 & \text{ otherwise} .\end{cases}.$$ where round $(x)$ denotes the round function. Compute the expected value of $g_n$ if $n$ is an integer chosen from interval $[1, 2019^2]$.

Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$ [Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$.

Define $a_n$: the coefficient of then item $x$ in $(3-\sqrt{x})^n$, where $n$ is a positive integer. Then $\lim_{n\to\infty}\left(\frac{3^2}{a_2}+\frac{3^3}{a_3}+\cdots+\frac{3^n}{a_n}\right)=$________.

A function $f$, defined on the set of real numbers $\mathbb{R}$ is said to have a [i]horizontal chord[/i] of length $a>0$ if there is a real number $x$ such that $f(a+x)=f(x)$. Show that the cubic $$f(x)=x^3-x\quad \quad \quad \quad (x\in \mathbb{R})$$ has a horizontal chord of length $a$ if, and only if, $0<a\le 2$.

Let $S = \{2, 3, 4, \ldots\}$ denote the set of integers that are greater than or equal to $2$. Does there exist a function $f : S \to S$ such that \[f (a)f (b) = f (a^2 b^2 )\text{ for all }a, b \in S\text{ with }a \ne b?\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.

Let $ P$ be the product of all non-zero digits of the positive integer $ n$. For example, $ P(4) \equal{} 4$, $ P(50) \equal{} 5$, $ P(123) \equal{} 6$, $ P(2009) \equal{} 18$. Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).

Let $a, b,c$ be positive real numbers. Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.

The graph of $ x^2 \plus{} y \equal{} 10$ and the graph of $ x \plus{} y \equal{} 10$ meet in two points. The distance between these two points is: $ \textbf{(A)}\ \text{less than 1} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ \sqrt{2}\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ \text{more than 2}$

Let $Q$ be a unit square in the plane: $Q = [0, 1] \times [0, 1]$. Let $T :Q \longrightarrow Q$ be defined as follows: \[T(x, y) =\begin{cases} (2x, \frac{y}{2}) &\mbox{ if } 0 \le x \le \frac{1}{2};\\(2x - 1, \frac{y}{2}+ \frac{1}{2})&\mbox{ if } \frac{1}{2} < x \le 1.\end{cases}\] Show that for every disk $D \subset Q$ there exists an integer $n > 0$ such that $T^n(D) \cap D \neq \emptyset.$

Prove the complex inequality $ |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|. $