Find all $3$-tuples of positive reals $(a,b,c)$ such that $$\begin{cases}a\sqrt[2019]b-c=a\\b\sqrt[2019]c-a=b\\c\sqrt[2019]a-b=c\end{cases}$$

Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality? [i](Walther Janous)[/i]

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$

Find all pairs of real numbers $(x,y)$ satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$

A function $g$ is such that for all integer $n$: $$g(n)=\begin{cases} 1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\ 0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 & \end{cases}$$ A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$: $$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$ $$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$ Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$

A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions: $\bullet$ the $n^2$ positive integers are pairwise distinct. $\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes. (a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$ (b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.

[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer $M$ there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least $M$ elements? [b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?

Prove that $\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.

Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties: (a) for any integer $n$, $f(n)$ is an integer; (b) the degree of $f(x)$ is less than $187$. Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns. [i]Proposed by YaWNeeT[/i]

$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$) Also we easily have $+,*$ in $\mathbb{N}_{10}$. first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.

Given points $A, B, C, D, E$, and $F$ on a line (not necessarily in that order) with $AB = 2$, $BC = 6$, $CD = 8$, $DE = 10$, $EF = 20$, and $FA = 22$. Find the distance between the two furthest points on the line.

Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the equation $$f(x^2+yf(x))=(1-x)f(y-x)$$ holds for all $x,y\in\mathbb{R}$.

Real numbers $x$ and $y$ satisfy the equations $x^2 - 12y = 17^2$ and $38x - y^2 = 2 \cdot 7^3$. Compute $x + y$.

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)

Let $ f(x) \equal{} x^3 \plus{} x \plus{} 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) \equal{} \minus{} 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.

a)Prove that for every n,natural number exist natural numbers a and b such that $(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$ b)Using first equation prove that for every n exist m such that $(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$

Does every positive polynomial in two real variables attain its lower bound in the plane?

Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$. Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.