A point $M$ inside triangle $ABC$ is such that $AM=AB/2$ and $CM=BC/2$. Points $C_0$ and $A_0$ lying on $AB$ and $CB$ respectively are such that $BC_0:AC_0 = BA_0:CA_0 = 3$. Prove that the distances from $M$ to $C_0$ and $A_0$ are equal.

Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$

Given a positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$ and $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are relatively prime to $n$. Find all $n$ for which one of the three numbers $n,\tau(n), \varphi(n)$ is the arithmetic mean of the other two.

If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and $x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008$, find the product $xyz$.

The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$. Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$.

Let $a,b$ be coprime positive integers with $a$ even and $a>b$. Show that there exist infinitely many pairs $(m,n)$ of coprime positive integers such that $m\mid a^{n-1}-b^{n-1}$ and $n\mid a^{m-1}-b^{m-1}$.

A number is chosen uniformly at random from the set of all positive integers with at least two digits, none of which are repeated. Find the probability that the number is even.

Let $n\ge3$ and $a_1,a_2,...,a_n \in \mathbb{R^{+}}$, such that $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1$. Prove that: $$a_1a_2...a_n \ge (n-1)^{\frac n4}$$

Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$. Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.

Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

Let $p$ be a prime. Suppose the mean of the nonzero quadratic residues mod $p$ is less than $\frac{p}{2}$. Show that the median of the nonzero quadratic residues mod $p$ is less than $\frac{p}{2}$.

There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]

Find the maximum number $ c$ such that for all $n \in \mathbb{N}$ to have \[ \{n \cdot \sqrt{2}\} \geq \frac{c}{n}\] where $ \{n \cdot \sqrt{2}\} \equal{} n \cdot \sqrt{2} \minus{} [n \cdot \sqrt{2}]$ and $ [x]$ is the integer part of $ x.$ Determine for this number $ c,$ all $ n \in \mathbb{N}$ for which $ \{n \cdot \sqrt{2}\} \equal{} \frac{c}{n}.$

Denote $g(k)$ as the greatest odd divisor of $k$. Put $f(k) = \dfrac{k}{2} + \dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1, x_2, x_3, ...$ by $x_1 = 1$, $x_{n+1} = f(x_n)$. Find $n$ such that $x_n = 800$.

Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

a) Given a quartet of positive numbers $(a,b,c,d)$. It is transformed to the new one according to the rule: $a'=ab, b' =bc, c'=cd,d'=da$. The second one is transformed to the third according to the same rule and so on. Prove that if at least one initial number does not equal 1, than You can never obtain the initial set. b) Given a set of $2^k$ ($k$-th power of two) numbers, equal either to $1$ or to $-1$. It is transformed as that was in the a) problem (each one is multiplied by the next, and the last -- by the first. Prove that You will always finally obtain the set of positive units.

Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.

Let $n \ge 2$ be an integer. Show that there exist $n+1$ numbers $x_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}$, so that $\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}$, where $\{ x \}$ is the fractionary part of $x$.

Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that \[\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}\]

Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$. [i]Proposed by Daniel Liu

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$. (2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$, and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$. Prove that the equation \[ a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w \] has at least one complex root with norm at most $3n|\omega|$.

Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.