In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And (i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$. (ii) $a \circ b \neq b \circ a$ when $a \neq b$. Prove that: a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$. b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

Show that for real numbers $x_1, x_2, ... , x_n$ we have: \[ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0 \] When do we have equality?

For any positive integer $n$, define $f(n)$ to be the smallest positive integer that does not divide $n$. For example, $f(1)=2$, $f(6)=4$. Prove that for any positive integer $n$, either $f(f(n))$ or $f(f(f(n)))$ must be equal to $2$.

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$. [asy] pair A, B, C, D; A = origin; real Bcoord = 3*sqrt(2) + sqrt(6); B = Bcoord/2*dir(180); C = sqrt(6)*dir(120); draw(A--B--C--cycle); D = (C-B)/2.4 + B; draw(A--D); label("$A$", A, dir(0)); label("$B$", B, dir(180)); label("$C$", C, dir(110)); label("$D$", D, dir(130)); [/asy] $ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$

Which of the following is equivalent to $ \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}}$ when $ x < 0$? $ \textbf{(A) } \minus{} x \qquad \textbf{(B) } x \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \sqrt {\frac x2} \qquad \textbf{(E) } x\sqrt { \minus{} 1}$

Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$ . [hide=Original]$$f(x+f(y))+f(x+y)=2x+f(y)+y$$[/hide]

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

The function $f : (0,1) \to R$ is defined by $f(x) = x$ if $x$ is irrational, $f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$. Find the maximum value of $f$ on the interval $(7/8,8/9)$.

A particle moves along the $x$-axis. It collides elastically head-on with an identical particle initially at rest. Which of the following graphs could illustrate the momentum of each particle as a function of time? [asy] size(400); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,5)); draw((0,1.5)--(5,1.5)); label("$p$",(0,5),N); label("$t$",(5,1.5),E); label("$\mathbf{(A)}$",(2.5,-0.5)); draw((0,1.5)--(2.5,1.5)--(2.5,0.75)--(4,0.75),black+linewidth(2)); draw((0,3.5)--(2.5,3.5)--(2.5,4.25)--(4,4.25),black+linewidth(2)); draw((8,0)--(8,5)); draw((8,1.5)--(13,1.5)); label("$p$",(8,5),N); label("$t$",(13,1.5),E); label("$\mathbf{(B)}$",(10.5,-0.5)); draw((8,1.5)--(10.5,1.5)--(10.5,2.5)--(12,2.5),black+linewidth(2)); draw((8,3.5)--(10.5,3.5)--(10.5,4.5)--(12,4.5),black+linewidth(2)); draw((16,0)--(16,5)); draw((16,1.5)--(21,1.5)); label("$p$",(16,5),N); label("$t$",(21,1.5),E); label("$\mathbf{(C)}$",(18.5,-0.5)); draw((16,1.5)--(18.5,1.5)--(18.5,2.25)--(20,2.25),black+linewidth(2)); draw((16,3.5)--(18.5,3.5)--(18.5,2.75)--(20,2.75),black+linewidth(2)); draw((24,0)--(24,5)); draw((24,1.5)--(29,1.5)); label("$p$",(24,5),N); label("$t$",(29,1.5),E); label("$\mathbf{(D)}$",(26.5,-0.5)); draw((24,1.5)--(26.5,1.5)--(26.75,3.25)--(28,3.25),black+linewidth(2)); draw((24,3.25)--(26.5,3.25)--(26.75,1.5)--(28,1.5),black+linewidth(2)); draw((32,0)--(32,5)); draw((32,1.5)--(37,1.5)); label("$p$",(32,5),N); label("$t$",(37,1.5),E); label("$\mathbf{(E)}$",(34.5,-0.5)); draw((32,1.5)--(34.5,1.5)--(34.5,0.5)--(36,0.5),black+linewidth(2)); draw((32,3.5)--(34.5,3.5)--(34.5,2.75)--(36,2.75),black+linewidth(2)); [/asy]

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $\mathbb N_0$ be the set of non-negative integers. There is a triple $(f,a,b)$, where $f$ is a function from $\mathbb N_0$ to $\mathbb N_0$ and $a,b\in\mathbb N_0$ that satisfies the following conditions: [list] [*]$f(1)=2$ [*]$f(a)+f(b)\leq 2\sqrt{f(a)}$ [*]For all $n>0$, we have $f(n)=f(n-1)f(b)+2n-f(b)$ [/list] Find the sum of all possible values of $f(b+100)$.