Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

5. Let $f : R \to R$ be a function that satisfies for all $x \in R$ (i) $| f(x)| \le 1$, and (ii) $f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$ Prove that $f$ is a periodic function

[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality $$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$ are periodic. [b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality $$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$ are periodic.

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

Let $\mathbb R$ be the real numbers. Set $S = \{1, -1\}$ and define a function $\operatorname{sign} : \mathbb R \to S$ by \[ \operatorname{sign} (x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x < 0. \end{cases} \] Fix an odd integer $n$. Determine whether one can find $n^2+n$ real numbers $a_{ij}, b_i \in S$ (here $1 \le i, j \le n$) with the following property: Suppose we take any choice of $x_1, x_2, \dots, x_n \in S$ and consider the values \begin{align*} y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right), \quad \forall 1 \le i \le n; \\ z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right) \end{align*} Then $z=x_1 x_2 \dots x_n$.

Solve the system \[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\] in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.

Prove the following inequality: $x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$ where $\forall _{x_i} x_i > 0$

Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.

Which natural numbers cannot be presented in that way: $[n+\sqrt{n}+\frac{1}{2}]$, $n\in\mathbb{N}$ $[y]$ is the greatest integer function.

Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence $$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$ is convergent and calculate its limit.

Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

Fin the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that: $$ \frac{f(x+y)+f(x)}{2x+f(y)} =\frac{2y+f(x)}{f(x+y)+f(y)} ,\quad\forall x,y\in\mathbb{N} . $$

Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$.

Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory. a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane. Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then $$g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1 \,\,\,\, (*)$$ Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property. b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$. c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$. d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$. e) Prove that for all $x , f(x) = g(x)$.

Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$

For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases: (i) $10 \leq n \leq 99$ (ii) $100 \leq n \leq 999$ (iii) $1000 \leq n \leq 9999$ (iv) $10000 \leq n \leq 99999$

Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that \[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]

Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$. [i]Marius Cavachi[/i]

Find all continous functions $f:[0,1]\to[0,2]$ with the property that $\left(\int_{\frac1{n+1}}^{\frac1n}xf(x)dx\right)^2=\int_{\frac1{n+1}}^{\frac1n}x^2f(x)dx, \forall n\in\mathbb N^*$. [i]Gabriel Marsanu, Andrei Nedelcu[/i]

If the function $f$ is defined by \[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is $\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

Let $ f: (0,\infty)\longrightarrow (0,\infty) $ a non-constant function having the property that $ f\left( x^y\right) = \left( f(x)\right)^{f(y)},\quad\forall x,y>0. $ Show that $ f(xy)=f(x)f(y) $ and $ f(x+y)=f(x)+f(y), $ for all $ x,y>0. $

Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]