Given the function $$y =|x^2 - 4x + 3|.$$ Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied: (1) $ f(rs) \equal{} f(r)\plus{}f(s)$ (2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3. (3) $ f(10) \equal{} 0$. Find $f(1985)$. Justify your answer.

Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]

Prove that if $ E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $ u < 1/2$, a point $ x\equal{}x(u)$ can be found so that \[ |(x\minus{}h,x\plus{}h) \cap E| \geq uh\] and \[ |(x\minus{}h,x\plus{}h) \cap (\mathbb{R} \setminus E)| \geq uh\] for all sufficiently small positive values of $ h$. [i]K. I. Koljada[/i]

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x $$ holds for all $ x,y \in \mathbb{R} $

If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$ [u]Babis[/u] [b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]

Function $f(n), n \in \mathbb N$, is defined as follows: Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$

Define the function $m$ of the three real variables $x$, $y$, $z$ by $m$($x$,$y$,$z$) = max($x^2$,$y^2$,$z^2$), $x$, $y$, $z$ ∈ $R$. Determine, with proof, the minimum value of $m$ if $x$,$y$,$z$ vary in $R$ subject to the following restrictions: $x$ + $y$ + $z$ = 0, $x^2$ + $y^2$ + $z^2$ = 1.

A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

There exists a function $G(x)$ such that $G(x)+G\left(\frac{x-\sqrt{3}}{x\sqrt{3}+1}\right)=\sqrt{2}+x$. Find $G(x)$. [i]Proposed by beanielove2.[/i]

Let \( n \) be a positive integer. A function \( f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\} \) is called \( n \)-Bolivian if it satisfies the following conditions: • \( f(0) = 0 \); • \( f(t) \in \{ t-1, f(t-1), f(f(t-1)), \dots \} \) for all \( t = 1, 2, \dots, n \). For example, if \( n = 3 \), then the function defined by \( f(0) = f(1) = 0 \), \( f(2) = f(3) = 1 \) is 3-Bolivian, but the function defined by \( f(0) = f(1) = f(2) = 0 \), \( f(3) = 1 \) is not 3-Bolivian. For a fixed positive integer \( n \), Gollum selects an \( n \)-Bolivian function. Smeagol, knowing that \( f \) is \( n \)-Bolivian, tries to figure out which function was chosen by asking questions of the type: \[ \text{How many integers } a \text{ are there such that } f(a) = b? \] given a \( b \) of his choice. Show that if Gollum always answers correctly, Smeagol can determine \( f \) and find the minimum number of questions he needs to ask, considering all possible choices of \( f \).

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, \[\left[f(x^2+1)\right]^{\sqrt{x}}=k;\] then, for every positive number $y$, \[\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}\] is equal to $\textbf{(A) }\sqrt{k}\qquad\textbf{(B) }2k\qquad\textbf{(C) }k\sqrt{k}\qquad\textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$

$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.

We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$. $\textit {Proposed by Ilija Jovcheski}$

Find all $a\in\mathbb R$ for which there exists a function $f\colon\mathbb R\rightarrow\mathbb R$, such that (i) $f(f(x))=f(x)+x$, for all $x\in\mathbb R$, (ii) $f(f(x)-x)=f(x)+ax$, for all $x\in\mathbb R$.

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations \[ \begin{array}{c} y=2[x]+3, \\ y=3[x-2]+5. \end{array} \]If $x$ is not an integer, then $x+y$ is $\textbf {(A) } \text{an integer} \qquad \textbf {(B) } \text{between 4 and 5} \qquad \textbf {(C) } \text{between -4 and 4} \qquad \textbf {(D) } \text{between 15 and 16} \qquad \textbf {(E) } 16.5$

Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\] in which $a,~ b,~ c$, and $d$ vary over the set of positive integers. (Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$