Prove that there are no $x\in\mathbb{R}^+$ such that $$x^{\lfloor x \rfloor }=\frac92.$$

For $a,b,c>0$, prove that (i) $\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0$ (ii) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$

Let $n,m$ be positive integers such that $n<m$ and $a_1, a_2, ..., a_m$ be different real numbers. (a) Find all polynomials $P$ with real coefficients and degree at most $n$ such that: $|P(a_i)-P(a_j)|=|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$. (b) If $n,m\ge 2$ does there exist a polynomial $Q$ with real coefficients and degree $n$ such that: $|Q(a_i)-Q(a_j)|<|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$ Edit: See #3

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers $a_{-1},\ldots,a_{-k}$ on a blackboard, so he decided to do the following process just for fun: if there are real numbers $a_{-k},\ldots,a_{n-1}$ on the blackboard, then he computes the polynomial $$P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).$$ He then writes a real number $a_n$, where $$a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.$$ If $a_n$ is undefined (that is, $P_n(i)+P_n(-i)=0$), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists $m>n\ge0$ such that $am=an$), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)

Let $a, b, c$ be nonnegative real numbers satisfying $a + b + c =3$. a) If $c > \frac32$, prove that $3(ab + bc + ca) - 2abc < 7$. b) Find the greatest possible value of $M =3(ab + bc + ca) - 2abc $.

Point $A,B$ are marked on the right branch of the hyperbola $y=\frac{1}{x},x>0$. The straight line $l$ passing through the origin $O$ is perpendicular to $AB$ and meets $AB$ and given branch of the hyperbola at points $D$ and $C$ respectively. The circle through $A,B,C$ meets $l$ at $F$. Find $OD:CF$

The product of the positive numbers $a, b, c, d$ and $e$ is equal to $1$. Prove that $$ \frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .$$

Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that $\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.

Find the largest integer $k$ with the following property: Whenever real numbers $x_1,x_2,\dots,x_{2024}$ satisfy \[x_1^2=(x_1+x_2)^2=\dots=(x_1+x_2+\dots+x_{2024})^2,\] at least $k$ of them are equal.

Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.

The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.

A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.

Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$, such that $$af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0$$ for all real $x, y, z$. (E. Barabanov)

Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.

The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and $$a_{n+2} = 2a_{n+1} + 41a_n$$Prove that $a_{2016}$ is divisible by $2017.$

A sequence of real numbers $(a_k)_{k \ge 0}$ is called [i]log-concave[/i] if for every $k \ge 1$, the inequality $a_{k - 1}a_{k + 1} \le a_k^2$ holds. Let $n, l \in \mathbb{N}$. Prove that the sequence $(a_k)_{k \ge 0}$ with general term \[a_k = \sum_{i = k}^{k + l} {n \choose i}\] is log-concave. Proposed by [i]Svetlana Poznanovikj[/i]

The sequence $ (a_{n})_{n\geq 1}$ is defined by $ a_{1} \equal{} 1, a_{2} \equal{} 3$, and $ a_{n \plus{} 2} \equal{} (n \plus{} 3)a_{n \plus{} 1} \minus{} (n \plus{} 2)a_{n}, \forall n \in \mathbb{N}$. Find all values of $ n$ for which $ a_{n}$ is divisible by $ 11$.

Let $a, b, c$ and $d$ be real numbers such that $$a^3+b^3+c^3+d^3=a+b+c+d=0$$ Prove that the sum of a pair of these numbers is equal to $0$. (LD Kurliandchik)

For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{} m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\] [i]V. T. Sos[/i]

Ivan and Krešo started to travel from Crikvenica to Kraljevica, whose distance is $15$ km, and at the same time Marko started from Kraljevica to Crikvenica. Each of them can go either walking at a speed of $5$ km/h, or by bicycle with the speed of $15$ km/h. Ivan started walking, and Krešo was driving a bicycle until meeting Marko. Then Krešo gave the bicycle to Marko and continued walking to Kraljevica, while Marko continued to Crikvenica by bicycle. When Marko met Ivan, he gave him the bicycle and continued on foot, so Ivan arrived at Kraljevica by bicycle. Find, for each of them, the time he spent in travel as well as the time spent in walking.

The real numbers $a,b$ and $c$ satisfy the inequalities $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$. Prove that $a+b+c=0$.

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

Transform the equation $$ab^2 \left(\frac{1}{(a + c)^2} +\frac{1}{(a- c)^2} \right) = (a -b)$$ into a simpler form, given that $a > c \ge 0$, $b > 0$.

Let $f: \mathbb{R}^{+}\rightarrow \mathbb{R}+$ be an increasing function. For each $u\in\mathbb{R}^{+}$, we denote $g(u)=\inf\{ f(t)+u/t \mid t>0\}$. Prove that: $(a)$ If $x\leq g(xy)$, then $x\leq 2f(2y)$; $(b)$ If $x\leq f(y)$, then $x\leq 2g(xy)$.