Let $ n > 1$ be a fixed integer. Define functions $ f_0(x) \equal{} 0,$ $ f_1(x) \equal{} 1 \minus{} \cos(x),$ and for $ k > 0,$ \[ f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x).\] If $ F(x) \equal{} \sum^n_{r\equal{}1} f_r(x),$ prove that [b](a)[/b] $ 0 < F(x) < 1$ for $ 0 < x < \frac{\pi}{n\plus{}1},$ and [b](b)[/b] $ F(x) > 1$ for $ \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.$

$f$ is a continuous real-valued function such that $f(x+y)=f(x)f(y)$ for all real $x$, $y$. If $f(2)=5$, find $f(5)$.

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold: $\qquad\ (1) \; f(1) = 1.$ $\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$ $\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$ $\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$ Find the sum of all possible values of $f(2023)$.

Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.

Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation \[(y+1)f(x+y) = f\left(xf(y)\right)\] For all non-negative real numbers $x$ and $y.$

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

[b]a)[/b] prove that the function $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ that is defined on the area $Re(s)>1$, is an analytic function. [b]b)[/b] prove that the function $\zeta(s)-\frac{1}{s-1}$ can be spanned to an analytic function over $\mathbb C$. [b]c)[/b] using the span of part [b]b[/b] show that $\zeta(1-n)=-\frac{B_n}{n}$ that $B_n$ is the $n$th bernoli number that is defined by generating function $\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}$.

Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]

Is there a real interval $ I $ for which there exists a primitivable function $ f:I\longrightarrow I $ with the property that $ (f\circ f) (x)=-x, $ for all $ x\in I $ ?

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying \[f(x+f(y))=x+f(f(y))\] for all real numbers $x$ and $y$, with the additional constraint $f(2004)=2005$.

Zeroes and ones are arranged in all the squares of $n\times n$ table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical. [i]Proposed by O. Vanyushina[/i]

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy \[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2}) \] for all $ x, y \in\mathbb R.$

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.