If $1+2x+3x^2 +...=9$, find $x$.

Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that $$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$ Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.

Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$. Show that $f$ is periodic.

Let $ f(x) \equal{} \log_{10} (\sin (\pi x)\cdot\sin (2\pi x)\cdot\sin (3\pi x) \cdots \sin (8\pi x))$. The intersection of the domain of $ f(x)$ with the interval $ [0,1]$ is a union of $ n$ disjoint open intervals. What is $ n$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36$

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that \[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\] Let $g$ be a function for which \[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\] Prove that $g$ is bounded.

For a given $\alpha >0$ let us consider the regular, non-vanishing $f(z)$ maps on the unit disc $\{ |z|<1 \}$ for which $f(0)=1$ and $\mathrm{Re}\, z\frac{f'(z)}{f(z)}>-\alpha$ ($|z|<1$). Show that the range of $$g(z)=\frac{1}{(1-z)^{2\alpha}}$$ contains the range of all other such functions. Here we consider that regular branch of $g(z)$ for which $g(0)=1$. (translated by Miklós Maróti)

Let $P$ be the set of points in the plane and $D$ the set of lines in the plane. Determine whether there exists a bijective function $f: P \rightarrow D$ such that for any three collinear points $A$, $B$, $C$, the lines $f(A)$, $f(B)$, $f(C)$ are either parallel or concurrent. [i]Gefry Barad[/i]

The graph of a function $f: \mathbb{R}\to\mathbb{R}$ has two has at least two centres of symmetry. Prove that $f$ can be represented as sum of a linear and periodic funtion.

Higher Secondary P2 Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)\geq 0\ \forall \ x\in \mathbb{R}$, $f'(x)$ exists $\forall \ x\in \mathbb{R}$ and $f'(x)\geq 0\ \forall \ x\in \mathbb{R}$ and $f(n)=0\ \forall \ n\in \mathbb{Z}$

Find all functions $f : \mathbb N \to \mathbb N$ such that: i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$, ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and iii) $f(m)=1$ if and only if $m=1$.

Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties: $\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$, $\bullet$ $f(30) = 1$, and $\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$. Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?

Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows: (i) $A(0, m, n) = m'$ for all $m, n \in N$ (ii) $A(k', 0, n) =\left\{ \begin{array}{ll} n & if \, \, k = 0 \\ 0 & if \, \,k = 1, \\ 1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$ (iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$. Compute $A(5, 3, 2)$. (H. Nestra)

Let be a function $ \xi:\mathbb{R}\to\mathbb{R} $ of class $ C^{\infty } $ such that $ \left| \frac{d^n\xi }{dx^n} \left( x_0 \right) \right|\le 1=\frac{d\xi}{dx}(0) , $ for any real numbers $ x_0, $ and all natural numbers $ n, $ and let be the function $ h:\mathbb{C}\longrightarrow\mathbb{C} , h(z)=1+\sum_{n\in\mathbb{N}} \left(\frac{z^n}{n!}\cdot\frac{d^n\xi }{dx^n} \left( 0 \right)\right) . $ [b]a)[/b] Show that $ h $ is well-defined and analytic. [b]b)[/b] Prove that $ h\bigg|_{\mathbb{R}} =\xi\bigg|_{\mathbb{R}} . $ [b]c)[/b] Demonstrate that $$ \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =4\sum_{p\in\mathbb{Z}}\frac{(-1)^p\xi\left( \frac{(1+2p)\pi}{2} \right)}{\left( (1+2p)\pi -2t_0\right)^2} , $$ for any $ t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) $ and that $$ \sum_{p\in\mathbb{Z}} \frac{(-1)^p\left(\xi\left( \frac{(1+2p)\pi}{2} \right)\right)^2}{1+2p} =\frac{\pi }{2} . $$ [b]d)[/b] Deduce that $ \xi\left( \frac{(1+2p)\pi}{2} \right)=(-1)^p, $ for any integer $ p, $ and that $$ \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =\frac{d}{dt}\left( \frac{\sin }{\cos} \right)\left( t_0 \right) , $$ for any $ t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) . $ [b]e)[/b] Conclude that $ \xi\bigg|_\mathbb{R} =\sin\bigg|_\mathbb{R} . $

If $ f(x) \equal{} 5x^2 \minus{} 2x \minus{} 1$, then $ f(x \plus{} h) \minus{} f(x)$ equals: $ \textbf{(A)}\ 5h^2 \minus{} 2h \qquad \textbf{(B)}\ 10xh \minus{} 4x \plus{} 2 \qquad \textbf{(C)}\ 10xh \minus{} 2x \minus{} 2 \\ \textbf{(D)}\ h(10x \plus{} 5h \minus{} 2) \qquad \textbf{(E)}\ 3h$

For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as \[ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . \] Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of \[ \sum_{n=1}^{11} (-1)^{n+1} a_n . \]

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.

Determine the value of $142857 + 285714 + 428571 + 571428.$ [i]Proposed by Ray Li[/i]

A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$

Let $r_1$, $r_2$, $\ldots$, $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$. Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$. Determine the minimum and maximum values of $f(n)$. Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$.

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]