$f(x)$ is a function defined on $\mathbb{R}$. $f(1)=1$, and for all $x\in\mathbb{R}$, $f(x+5)\geq x+5,f(x+1)\leq f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$________.

Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions. [list=disc] [*] $f(a)$ is not an integer for some rational number $a$. [*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers. [/list]

Let $ f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $ \{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $ x$. Prove that $ f$ is nondecreasing.

Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}\plus{}b^{2}\plus{}c^{2}\equal{}1$ prove the inequality: \[ \frac{a^{5}\plus{}b^{5}}{ab(a\plus{}b)}\plus{} \frac {b^{5}\plus{}c^{5}}{bc(b\plus{}c)}\plus{}\frac {c^{5}\plus{}a^{5}}{ca(a\plus{}b)}\geq 3(ab\plus{}bc\plus{}ca)\minus{}2.\]

Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

Let \( \mathbb{N} \) be the set of all positive integers. We say that a function \( f: \mathbb{N} \to \mathbb{N} \) is Georgian if \( f(1) = 1 \) and, for every positive integer \( n \), there exists a positive integer \( k \) such that \[ f^{(k)}(n) = 1, \quad \text{where } f^{(k)} = f \circ f \cdots \circ f \quad \text{(applied } k \text{ times)}. \] If \( f \) is a Georgian function, we define, for each positive integer \( n \), \( \text{ord}(n) \) as the smallest positive integer \( m \) such that \( f^{(m)}(n) = 1 \). Determine all positive real numbers \( c \) for which there exists a Georgian function such that, for every positive integer \( n \geq 2024 \), it holds that \( \text{ord}(n) \geq cn - 1 \).

Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear

Find the first degree polynomial function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equation $$ f(x-1)=-3x-5-f(2), $$ for all real numbers $ x. $

For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist? [i](A. Golovanov)[/i]

Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?

Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\] for all $x,y\in\mathbb{R}$

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $ [b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $ [b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.

[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that \[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\] for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]

[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]

Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers such that \[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \] for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$. [i]David Yang[/i]

Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.

Compute the value of $$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$ Your answer should not involve any trigonometric functions. [i]Proposed by Howard Halim[/i]

Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders