The set of all real numbers $ x$ for which \[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \]is defined is $ \{x|x > c\}$. What is the value of $ c$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$

For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.

4. Let $\mathbb{N}$ denote the strictly positive integers. A function $f$ : $\mathbb{N}$ $\to$ $\mathbb{N}$ has the following properties which hold for all $n \in$ $\mathbb{N}$: a) $f(n)$ < $f(n+1)$; b) $f(f(f(n)))$ = 4$n$ Find $f(2022)$.

Function $f(x)=-\frac{1}{2}x^2+\frac{13}{2}$. If the minumum and maximum value of $f(x)$ are $2a$ and $2b$ respectively on $[a,b]$. Find $a,b$.

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$

If $a,b,c$ are three real numbers and \[ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t \] for some real number $t$, prove that $abc + t = 0 .$

Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?

Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y),$ such that $f(0,k)=f(k,0)=2^k$ and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers $a, b.$ Determine the number of positive integers $n\leq2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b)=n$. [i]Proposed by Michael Ren[/i]

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy \[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]

Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition $f(f(x)+x+y)=2x+f(y)$ for any real $x$ and $y$.

Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$.

Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

Let $f(x)$ be an even twice differentiable function such that $f''(0)\ne0$. Prove that $f(x)$ has a local extremum at $x=0$.

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.

Let $u(t)$ be a continuous function in the system of differential equations $$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$ Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$ at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value $t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$

Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies \[ \sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3}, \] compute the ordered triple $(a, b, c)$.

A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon. Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that $$ (f(\sin x))^2-3f(x)=-2, $$ for any real numbers $ x. $ [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

Let $\mathbb{N}_0=\{0,1,2 \cdots \}$. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that: \[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 \] where we define: $f^1(n)=f(n)$ and $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that (a) $f(1)=1$ (b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$. (c) $f(n)$ divides $f(m)$ if $m>n$.