Let $\mathbb Q$ be the set of all rational numbers and $\mathbb R$ be the set of real numbers. Function $f: \mathbb Q \to \mathbb R$ satisfies the following conditions: (i) $f(0) = 0$, and for any nonzero $a \in Q, f(a) > 0.$ (ii) $f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q.$ (iii) $f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.$ Let $x$ be an integer and $f(x) \neq 1$. Prove that $f(1 + x + x^2+ \cdots + x^n) = 1$ for any positive integer $n.$

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

Range of function $f(x)=\arctan x+\frac{1}{2}\arcsin x$ is $\text{(A)}(-\pi,\pi)\qquad\text{(B)}[-\frac{3}{4}\pi,\frac{3}{4}\pi]\qquad\text{(C)}(-\frac{3}{4}\pi,\frac{3}{4}\pi)\qquad\text{(D)}[-\frac{1}{2}\pi,\frac{1}{2}\pi]$

Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.

Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that: a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$. b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.

Let $A_n$ be the set of all the sums $\displaystyle \sum_{k=1}^n \arcsin x_k $, where $n\geq 2$, $x_k \in [0,1]$, and $\displaystyle \sum^n_{k=1} x_k = 1$. a) Prove that $A_n$ is an interval. b) Let $a_n$ be the length of the interval $A_n$. Compute $\displaystyle \lim_{n\to \infty} a_n$.

The function $f(x) = x^2+ax+b$ has two distinct zeros. If $f(x^2+2x-1)=0$ has four distinct zeros $x_1<x_2<x_3<x_4$ that form an arithmetic sequence, compute the range of $a-b$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 11)[/i]

The image that maps $x$ to $1 - x$ is called [i]complement[/i], the image that maps $x$ to $\frac{1}{x}$ is called [i]invert[/i]. Two numbers $x$ and $y$ are called related if they can be transferred into each other by means of [i]complementation [/i]and/or [i]inversion[/i]. A [i]family [/i] is a collection of numbers where every two elements are related. Determine the maximum size $n$ of such a family. Show that the number line can be divided into $n$ parts, such that each of those $n$ parts contains exactly one number from each $n$-number family.

Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation $$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$ for all positive $x,y$, then $$2f(\sqrt{xy})=f(x)+f(y)$$ for every pair $x,y$ of positive numbers.

Let $f:\mathbb{N} \rightarrow \mathbb{N} \setminus \left \{ 1 \right \}$ be a function which satisfies both the inequality $f(a+f(a)) \leq 2a+3$ and the equation $$f(f(a)+b) = f(a+f(b)),$$ for any two $a,b \in \mathbb{N}$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be defined with: $g(a)$ is the largest prime divisor of $f(a)$. Prove that there exist integers $a>b>2024$ such that $b|a$ and $g(a) = g(b)$.

Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$. a) Solve the equation $ f(x) = 0$. b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]

Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $|f(k)| \leq k$ for all positive integers $k$ and there is a prime number $p>2024$ which satisfies both of the following conditions: $1)$ For all $a \in \mathbb{N}$ we have $af(a+p) = af(a)+pf(a),$ $2)$ For all $a \in \mathbb{N}$ we have $p|a^{\frac{p+1}{2}}-f(a).$ [i]Authored by Nikola Velov[/i]

$f(x)=\frac{4^x}{4^x+2}$, then $f(\frac{1}{1001})+f(\frac{2}{1001})+\cdots+f(\frac{1000}{1001})=$________.

Let $\mathbb N$ be the set of positive integers. Find all functions $f: \mathbb N \to \mathbb N$ that satisfy the equation \[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \] for all $a,b,c \ge 2$. (Here $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)

$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is [i][u]expanding[/u][/i] for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$ $\indent$ [b](a)[/b] Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ? $\indent$ [b](b)[/b] Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?

For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$

Prove that for $k>10$ Nazar may replace in the following product some one $\cos$ by $\sin$ so that the new function $f_{1}(x)$ would satisfy inequality $|f_{1}(x)|\le 3\cdot 2^{-1-k}$ for all real $x$. \[f(x) = \cos x \cos 2x \cos 3x \dots \cos 2^{k}x \] [i]N. Agakhanov[/i]

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that for any real numbers $x,y$ with $y\neq 0$ we have $$f(f(x)+y)f\left(\frac{1}{y}\right)=xf\left(\frac{1}{y}\right) + 1.$$ [i]Proposed by Marius Cerlat[/i]

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\] for all real numbers $x$ and $y$.

Let $f:\mathbb R\to(0,\infty)$ be a differentiabe function, and suppose that there exists a constant $L>0$ such that $$|f'(x)-f'(y)|\leq L|x-y|$$ for all $x,y$. Prove that $$(f'(x))^2<2Lf(x)$$ holds for all $x$.

Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free. [i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)

Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.

Consider two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the graph of the function $y = \frac{1}{x}$ such that $0 < x_1 < x_2$ and $AB = 2 \cdot OA$, where $O = (0, 0)$. Let $C$ be the midpoint of the segment $AB$. Prove that the angle between the $x$-axis and the ray $OA$ is equal to three times the angle between the $x$-axis and the ray $OC$.