Function $f:\mathbb{R}\to\mathbb{R}$, satisfies that $f(0)=1$, and $f(xy+1)=f(x)f(y)-f(y)-x+2$, then $f(x)=$________.

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following: $i)$ $f(1)=1$; $ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

We define $N$ as the set of natural numbers $n<10^6$ with the following property: There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$. Find $|N|$.

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following: [list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list] (a) Prove that the function $f$ is unique. (b) Find $f(\frac{\sqrt{5}-1}{2})$.

Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$ for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.

Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.

In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.

Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$. [i]Marian Andronache[/i]

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$: (a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$ where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$

In a triangle $ABC$, let $a,b,c$ be its sides and $m,n,p$ be the corresponding medians. For every $\alpha>0$, let $\lambda(\alpha)$ be the real number such that $$a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.$$ (a) Compute $\lambda(2)$. (b) Find the limit of $\lambda(\alpha)$ as $\alpha$ approaches $0$. (c) For which triangles $ABC$ is $\lambda(\alpha)$ independent of $\alpha$?

The function $f$ satisfies the functional equation \[f(x) +f(y) = f(x + y ) - xy - 1\] for every pair $x,~ y$ of real numbers. If $f( 1) = 1$, then the number of integers $n \neq 1$ for which $f ( n ) = n$ is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinite}$

On a board are drawn the points $A,B,C,D$. Yetti constructs the points $A^\prime,B^\prime,C^\prime,D^\prime$ in the following way: $A^\prime$ is the symmetric of $A$ with respect to $B$, $B^\prime$ is the symmetric of $B$ wrt $C$, $C^\prime$ is the symmetric of $C$ wrt $D$ and $D^\prime$ is the symmetric of $D$ wrt $A$. Suppose that Armpist erases the points $A,B,C,D$. Can Yetti rebuild them? $\star \, \, \star \, \, \star$ [b]Note.[/b] [i]Any similarity to real persons is purely accidental.[/i]

Given any two positive real numbers $x$ and $y$, then $x\Diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x\Diamond y$ satisfies the equations $(x\cdot y)\Diamond y=x(y\Diamond y)$ and $(x\Diamond 1)\Diamond x=x\Diamond 1$ for all $x,y>0$. Given that $1\Diamond 1=1$, find $19\Diamond 98$.

Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$

There are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity \[ \tan^{-1} \bigl( \frac{1}{x} - \frac{x}{8} \bigr) + \tan^{-1}(ax) + \tan^{-1}(bx) = \frac{\pi}{2} \, . \] (Throughout this equation, $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.) What is the value of $a^2 + b^2$?

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.