a) Find at least two functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ b) Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ Show that $ f(x^3)\geq x^2,$ for all $x \in \mathbb{R}^+.$ Can we find the best constant $a\in \Bbb{R}$ such that $f(x)\geq x^a,$ for all $x \in \mathbb{R}^+?$

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

Given $n \geq 2$ positive real numbers $x_1 \leq x_2 \leq \ldots \leq x_n$ satisfying the equalities $$x_1+x_2+\ldots+x_n=4n$$ $$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=n$$ Prove that $\frac{x_n}{x_1} \geq 7+4\sqrt{3}$

If $z$ is a complex number such that \[ z + z^{-1} = \sqrt{3}, \] what is the value of \[ z^{2010} + z^{-2010} \, ? \]

Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$. Find the term $a_{2013}$.

Prove that for all natural $ m$, $ n$ polynomial $ \sum_{i \equal{} 0}^{m}\binom{n\plus{}i}{n}\cdot x^i$ has at most one real root.

Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$

A polynomial $f (x)$ with real coefficients is called [i]completely reducible[/i] if it is a product of at least two non-constant polynomials whose coefficientsare all nonnegative real numbers. Show: If $f (x^{2011})$ is completely reducible, then $f(x)$ is also.

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Prove that there exists a uniqe $P(x)$ polynomial with real coefficients such that\\ $xy-x-y|(x+y)^{1000}-P(x)-P(y)$ for all real $x,y$.

Prove that $ \forall x\in \mathbb{R} $ , $ \cos ^7x+\cos ^7(x+\frac {2\pi}{3})+\cos ^7(x+\frac {4\pi}{3})=\frac {63}{64}\cos 3x $

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy $$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$ [i]Proposed by Kyle Lee[/i]

Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that $$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$ for all positive real numbers $x$ and $y$.

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real numbers $x , y$ this equality holds : $$f(yf(x)+f(x)f(y))=xf(y)+f(xy)$$ [i]Proposed by Navid Safaei[/i]

[b]a)[/b] Given two positive reals $ x,y, $ prove that $ \min\left( x,1/x+y,1/y \right)\le\sqrt 2. $ and determine when equality holds. [b]b)[/b] Find all triplets of real numbers $ (a,b,c) $ having the property that for every triplet of real numbers $ (x,y,z) , $ the following equality holds: $$ |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z| $$

For positive integer $n,$ let $a_n=\sum_{k=0}^{[\frac{n}{2}]}\binom{n-2}{k}(-\frac{1}{4})^k.$ Find $a_{1997}.$ (For real $x,$ $[x]$ is defined as largest integer that does not exceeds $x.$)

Let $\{a_n\}_{n\geq 0}$ be an arithmetic sequence with difference $d$ and $1\leq a_0\leq d$. Denote the sequence as $S_0$, and define $S_n$ recursively by two operations below: Step $1$: Denote the first number of $S_n$ as $b_n$, and remove $b_n$. Step $2$: Add $1$ to the first $b_n$ numbers to get $S_{n+1}$. Prove that there exists a constant $c$ such that $b_n=[ca_n]$ for all $n\geq 0$, where $[]$ is the floor function.

Let $f : \mathbb Q \to \mathbb Q$ be a function such that for any $x,y \in \mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide whether it follows that there exists a constant $c$ such that $f(x) - cx$ is an integer for every rational number $x$. [i]Proposed by Victor Wang[/i]

Show that there exists a continuos function $f: [0,1]\rightarrow [0,1]$ such that it has no periodic orbit of order $3$ but it has a periodic orbit of order $5$.

[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions. [b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale. [b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number? [b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it? [b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ [i]Proposed by Krit Boonsiriseth[/i]

Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$