[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.

Find all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that for all positive integers $m,n$ and $a$ we have a) $f(f(m)f(n))=mn$ and b) $f(2024a+1)=2024a+1$.

Given positive integers $n$, $P_1$, $P_2$, …$P_n$ and two sets \[B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \}\] A function $f:S \to \mathbb{Z}$ is called [b]Real[/b], if and only if for any positive integers $(y_1,y_2,…,y_n)$ and positive integer $a$ which satisfied $ 1 \leq y_i \leq P_i-a$ $\forall i \in \mathbb{N}$, we always have: \begin{align*} \sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\ \sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&. \end{align*} Find the minimum of $\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|$, where $f$ is a [b]Real[/b] function. [i]Proposed by tob8y[/i]

Two parabolas, $y = ax^2 + bx + c$ and $y = -ax^2- bx - c$, intersect at $x = 2$ and $x = -2$. If the $y$-intercepts of the two parabolas are exactly $2$ units apart from each other, compute $|a+b+c|$.

Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root. [i]Proposed by Hesam Rajabzadeh [/i]

Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality \[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\] holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as \[ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . \] Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of \[ \sum_{n=1}^{11} (-1)^{n+1} a_n . \]

Find all polynomials $f$ that satisfy the equation \[\frac{f(3x)}{f(x)} = \frac{729 (x-3)}{x-243}\] for infinitely many real values of $x$.

Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$. [i]Mircea Fianu[/i]

Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$, $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and $\vspace{0.1cm}$ $\hspace{1cm}ii) f(n)$ divides $n^3$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$

We represent the number line $R$ as the union of two non-empty sets $A, B$ different from $R$. Prove that one of the sets $A, B$ does not have the following property: the difference of any elements of the set belongs to the same set.

Let $x$, $y$, $z$ be real numbers such that $x+y+z \ne 0$. Find the minimum value of $\frac{|x|+|x+4y|+|y+7z|+2|z|}{|x+y+z|}$

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

Call a triple $(a,b,c)$ of positive numbers [i]mysterious [/i]if \[\sqrt{a^2+\frac{1}{a^2c^2}+2ab}+\sqrt{b^2+\frac{1}{b^2a^2}+2bc}+\sqrt{c^2+\frac{1}{c^2b^2}+2ca}=2(a+b+c).\] Prove that if the triple $(a,b,c)$ is mysterious, then so is the triple $(c,b,a)$. [i]Proposed by A. Kuznetsov, K. Sukhov[/i]

Find all positive real numbers $x, y, z$, such that $$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$

Given the disjoint finite sets of natural numbers $ A $ and $ B $, consisting of $ n $ and $ m $ elements, respectively. It is known that every natural number belonging to $ A $ or $ B $ satisfies at least one of the conditions $ k + 17 \in A $, $ k-31 \in B $. Prove that $ 17n = 31m $

Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$, $$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?

Find all function $f,g: \mathbb{Q} \to \mathbb{Q}$ such that \[\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array}\] for all $x,y \in \mathbb{Q}$.

[b]Problem 3[/b] Let $ P(x) = x^{2015} -2x^{2014}+1$ and $ Q(x) = x^{2015} -2x^{2014}-1$. Determine for each of the polynomials $P$ and $Q$ whether it is a divisor of some nonzero polynomial $c_0 + c_{1}x +\ldots + c_{n}x^n$ n whose coefficients $c_i$ are all in the set $ \{ -1, 1\}$.

Let $ (x_n)$ be the sequence of natural number such that: $ x_1\equal{}1$ and $ x_n<x_{n\plus{}1}\leq 2n$ for $ 1\leq n$. Prove that for every natural number $ k$, there exist the subscripts $ r$ and $ s$, such that $ x_r\minus{}x_s\equal{}k$.

Find all postitive integers n such that $$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$ where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.

The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation \[ x^4 + *x^3 + *x^2 + *x + * = 0 \] with a positive real number. Who has a winning strategy if the goals of the players are: a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots? b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?

Recall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \leq a<b \leq 100$, find the number of ordered pairs $(a, b)$ satisfying the following equation. $$[a+\frac{b}{a}]=[b+\frac{a}{b}]$$