Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.

Let $ n\left(A\right)$ be the number of distinct real solutions of the equation $ x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\left(A\right)$ is $\textbf{(A)}\ \left\{0,1,2,3,4,5,6\right\} \\ \textbf{(B)}\ \left\{0,2,4,6\right\} \\ \textbf{(C)}\ \left\{0,3,4,6\right\} \\ \textbf{(D)}\ \left\{0,2,3,4,6\right\} \\ \textbf{(E)}\ \left\{0,2,3,4\right\}$

Solve the following system of equations in real numbers: \[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]

$f(x)=ax^2-c$. If$-4\leq f(1)\leq -1,-z\leq f(2)\leq 5$, then $\text{(A)}7\leq f(3)\leq26\qquad\text{(B)}-4\leq f(3)\leq15\qquad\text{(C)}-1\leq f(3)\leq23\qquad\text{(D)}-\frac{28}{3}\leq f(3)\leq\frac{35}{3}$

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

Find all positive integers $n$ that have precisely $\sqrt{n+1}$ natural divisors.

Let $I \subset \mathbb{R}$ be a nondegenerate interval and $f:I \to \mathbb{R}$ a differentiable function. We denote $J= \left\{ \frac{f(b)-f(a)}{b-a} : a,b \in I, a<b \right\}.$ Prove that: $a)$ $J$ is an interval; $b)$ $J \subset f'(I),$ and the set $f'(I) \setminus J$ contains at most two elements; $c)$ Using parts $a)$ and $b),$ deduce that $f'$ has the intermediate value property.

Find both extrema of the function $ x\to\frac{\sin x-3}{\cos x +2} .$

Find all real numbers $ x$ such that: $ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$ (Here $ [x]$ denotes the largest integer not exceeding $ x$.)

Let $f:[0,1]\to [0,1]$ be a continuous function. We say $f\equiv 0$ if $f(x)=0$ for all $x\in [0,1]$ and similarly $f\not\equiv 0$ if there exists at least one $x\in [0,1]$ such that $f(x)\neq 0$. Suppose $f\not\equiv 0$, $f \circ f \not\equiv 0$ but $f \circ f \circ f \equiv 0$. Do there exists such an $f$? If yes construct such an function, if no prove it

A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$, $$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$, showing all necessary details in your derivation.

The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$? $ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$

Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality: \begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}

Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points, then at least at one point it is differentiable?

Real function $f$ [b]generates[/b] real function $g$ if there exists a natural $k$ such that $f^k=g$ and we show this by $f \rightarrow g$. In this question we are trying to find some properties for relation $\rightarrow$, for example it's trivial that if $f \rightarrow g$ and $g \rightarrow h$ then $f \rightarrow h$.(transitivity) (a) Give an example of two real functions $f,g$ such that $f\not = g$ ,$f\rightarrow g$ and $g\rightarrow f$. (b) Prove that for each real function $f$ there exists a finite number of real functions $g$ such that $f \rightarrow g$ and $g \rightarrow f$. (c) Does there exist a real function $g$ such that no function generates it, except for $g$ itself? (d) Does there exist a real function which generates both $x^3$ and $x^5$? (e) Prove that if a function generates two polynomials of degree 1 $P,Q$ then there exists a polynomial $R$ of degree 1 which generates $P$ and $Q$. Time allowed for this problem was 75 minutes.

Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

Prove that if m,n are nonnegative integers and 0<=x<=1 then $(1-x^n)^m + (1-(1-x)^m)^n \ge 1$

Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $ Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$

Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N. Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.

A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Let $f:\mathbb{Z}\rightarrow\{ 1,2,\ldots ,n\}$ be a function such that $f(x)\not= f(y)$, for all $x,y\in\mathbb{Z}$ such that $|x-y|\in\{2,3,5\}$. Prove that $n\ge 4$. [i]Ioan Tomescu[/i]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.