Let $m$ be an odd positive integer greater than $1$. Let $S_m$ be the set of all non-negative integers less than $m$ which are of the form $x+y$, where $xy-1$ is divisible by $m$. Let $f(m)$ be the number of elements of $S_m$. [b](a)[/b] Prove that $f(mn)=f(m)f(n)$ if $m$, $n$ are relatively prime odd integers greater than $1$. [b](b)[/b] Find a closed form for $f(p^k)$, where $k>0$ is an integer and $p$ is an odd prime.

Find all polynomials P(x) satisfying the equation $(2x-1)P(x) = (x-1)P(2x), \forall x.$

We have $105$ coins, among which we know that there are three fake ones. Authentic coins have all the same weight, which is greater than that of the false ones, which also have the same weight. Determine from can $26$ authentic coins be selected by weighing only two in one two pan balance.

Let $ABCD$ be a cyclic quadrilateral. Let $H_{1}$ be the orthocentre of triangle $ABC$. Point $A_{1}$ is the image of $A$ after reflection about $BH_{1}$. Point $B_{1}$ is the image of of $B$ after reflection about $AH_{1}$. Let $O_{1}$ be the circumcentre of $(A_{1}B_{1}H_{1})$. Let $H_{2}$ be the orthocentre of triangle $ABD$. Point $A_{2}$ is the image of $A$ after reflection about $BH_{2}$. Point $B_{2}$ is the image of of $B$ after reflection about $AH_{2}$. Let $O_{2}$ be the circumcentre of $(A_{2}B_{2}H_{2})$. Lets denote by $\ell_{AB}$ be the line through $O_{1}$ and $O_{2}$. $\ell_{AD}$ ,$\ell_{BC}$ ,$\ell_{CD}$ are defined analogously. Let $M=\ell_{AB} \cap \ell_{BC}$, $N=\ell_{BC} \cap \ell_{CD}$, $P=\ell_{CD} \cap \ell_{AD}$,$Q=\ell_{AD} \cap \ell_{AB}$. Prove that $MNPQ$ is cyclic.

For prime number $p$, let $f_p(x)=x^{p-1} +x^{p-2} + \cdots + x + 1$. (1) When $p$ divides $m$, prove that there exists a prime number that is coprime with $m(m-1)$ and divides $f_p(m)$. (2) Prove that there are infinitely many positive integers $n$ such that $pn+1$ is prime number.

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

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 $i = 1, \cdots , 2n$.

The function $f$ is defined by $$f(x) = \Big\lfloor \lvert x \rvert \Big\rfloor - \Big\lvert \lfloor x \rfloor \Big\rvert$$for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$? $\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\}$ $\textbf{(D) } \{0\} \qquad\textbf{(E) } \text{The set of nonnegative integers} $

Find all natural numbers such that each is equal to the sum of the factorials of its digits. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

Numbers $1, 2,\ldots, n$ are written on the board. By one move, we replace some two numbers $ a, b$ with the number $a^2-b{}$. Find all $n{}$ such that after $n-1$ moves it is possible to obtain $0$.

Let be a natural number $ n, $ a number $ t\in (0,1) $ and $ n+1 $ numbers $ a_0\ge a_1\ge a_2\ge\cdots\ge a_n\ge 0. $ Prove the following matrix inequality: $$ \begin{vmatrix}\frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0& 0 & \cdots & 0 & 0 \\ 0 & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 \\ a_0 & a_1 & a_2 & a_3 & \cdots & a_{n-1} & a_n \end{vmatrix}^2\le a_0^2\left( 1+\frac{1}{t^2} \right) $$

Show that there do not exist positive integers $x,y,z$ such that $$x^x + y^y = 9^z$$ [i]usjl[/i]

In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set.

Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur. (I.Bogdanov)

On the island of knights and liars, a tennis tournament was held, in which $100$ people participated in. Each two of them played exactly $1$ time with the other one. After the tournament, each of the participants declared: “I have beaten as many knights as liars,” while all the knights told the truth, and all the liars lied. What is the largest number of knights that could participate in the tournament?

Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$. Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $2g(\mu (a,b)) = g(a)+g(b)$. Decide which of the two mean values $m$ and $\mu$ is larger.

Let $p_0(x),p_1(x),p_2(x),\ldots$ be polynomials such that $p_0(x)=x$ and for all positive integers $n$, $\dfrac{d}{dx}p_n(x)=p_{n-1}(x)$. Define the function $p(x):[0,\infty)\to\mathbb{R}$ by $p(x)=p_n(x)$ for all $x\in [n,n+1)$. Given that $p(x)$ is continuous on $[0,\infty)$, compute \[\sum_{n=0}^\infty p_n(2009).\]

In a group of $20$ people, each person sends a letter to $10$ of the others. Prove that there are two persons who send a letter to each other.

A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums \[a_{k_1}\] \[a_{k_1}+a_{k_2}\] \[a_{k_1}+a_{k_2}+a_{k_3}\] \[\vdots\] \[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\] have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.

The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that \[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]