Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)

The sequence $\{x_n\}_{n=0}^\infty$ is defined by the following equations: $x_n=\sqrt{x_{n-1} x_{n-2}+\frac{n}{2}}$ ,$\forall$ $n\geq 2$, $x_0=x_1=1$. Prove that there exist a real number $a$, such that $an<x_n<an+1$ for each natural number $n$.

For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$. [i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$.

p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March $6$, $2013$ the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in $2014$ the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower? p2. In one season, the Indonesian Football League is participated by $20$ teams football. Each team competes with every other team twice. The result of each match is $3$ if you win, $ 1$ if you draw, and $0$ if you lose. Every week there are $10$ matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain? p3. Look at the following picture. The quadrilateral $ABCD$ is a cyclic. Given that $CF$ is perpendicular to $AF$, $CE$ is perpendicular to $BD$, and $CG$ is perpendicular to $AB$. Is the following statements true? Write down your reasons. $$\frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF}$$ [img]https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png[/img] p4. Suppose $M=2014^{2014}$. If the sum of all the numbers (digits) that make up the number $M$ equals $A$ and the sum of all the digits that make up the number $A$ equals $B$, then find the sum of all the numbers that make up $B$. p5. Find all positive integers $n < 200$ so that $n^2 + (n + 1)^2$ is square of an integer.

Three positive numbers are such that the sum of any one of them with the sum of squares of the remaining two numbers is the same. Is it true that all numbers are the same? (Author: L. Emelyanov)

Show that there are two real solutions to: $$\begin{cases} x + y^2 + z^4 = 0 \\ y + z^2 + x^4 = 0 \\ z + x^2 + y^5 = 0\end {cases}$$

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

Let $r>s$ be positive integers. Let $P(x)$ and $Q(x)$ be distinct polynomials with real coefficients, non-constant(s), such that $P(x)^r-P(x)^s=Q(x)^r-Q(x)^s$ for every $x\in \mathbb{R}$. Prove that $(r,s)=(2,1)$.

Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that : \[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\] [i]Proposed by Mohammad Ahmadi[/i]

Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$).

Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.) [i]Proposed by David Altizio[/i]

Determine whether or not there exists a sequence of integers $a_1,a_2,\dots, a_{19}, a_{20}$ such that, the sum of all the terms is negative, and the sum of any three consecutive terms is positive.

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

Find the coefficient of the fourth term of the expansion of $(x+y)^{15}$.

Prove for the sidelengths $a,b,c$ of a triangle $ABC$ the inequality $\frac{a^3}{b+c-a}+\frac{b^3}{c+a-b}+\frac{c^3}{a+b-c}\ge a^2+b^2+c^2$

The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and \[ 1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 + (p-1)x}}}=\frac{P(x)}{Q(x)}. \] Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$. [i]Andrew Gu[/i]

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

In maths class Albrecht had to compute $(a+2b-3)^2$ . His result was $a^2 +4b^2-9$ . ‘This is not correct’ said his teacher, ‘try substituting positive integers for $a$ and $b$.’ Albrecht did so, but his result proved to be correct. What numbers could he substitute? a) Show a good substitution. b) Give all the pairs that Albrecht could substitute and prove that there are no more.

Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.

Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.

Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.