Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$; what is the sum of all possible distances from $P$ to line $AB$?

We are given a triangle $ABC$ and two circles ($k_1$ and $k_2$) so the diameter of $k_1$ is $AB$ and the diameter of $k_2$ is $AC$. Let the intersection of $BC$ line segment and $k_1$ (that isn’t $B$) be $P,$ and the intersection of $BC$ line segment and $k_2$ (that isn’t $B$) be $Q$. We know, that $AB = 3003$ and $AC = 4004$ and $BC = 5005$. What is the distance between $P$ and $Q$?

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there are three players A,B, and C such that A beats B, B beats C, and C beats A. Such a triple of player is called a cycle. Determine the number of maximum cycles such a tournament can have.

If $a,b$ and $c$ are three (not necessarily different) numbers chosen randomly and with replacement from the set $\{1,2,3,4,5 \}$, the probability that $ab+c$ is even is $\textbf{(A)}\ \dfrac{2}{5} \qquad \textbf{(B)}\ \dfrac{59}{125} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{64}{125} \qquad \textbf{(E)}\ \dfrac{3}{5}$

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$. [i]Proposed by Athanasios Kontogeorgis, Greece[/i]

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that [list] [*]$m = 1$ and $l = 2k$; or [*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$. [/list]

Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$. [i]Proposed by Nathan Ramesh

Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$. (a) Prove that $\textrm{Im} \, f$ is finite or numerable. (b) Find all continuous functions which have the property mentioned in the hypothesis.

Let $n\geq 2$ be a positive integer. Prove that there exists a positive integer $m$, such that $n\mid m, \ m<n^4$ and at most four distinct digits are used in the decimal representation of $m$.

Let $g$ be a Fibonacci primitive root $\pmod{p}$. i.e. $g$ is a primitive root $\pmod{p}$ satisfying $g^2 \equiv g+1\; \pmod{p}$. Prove that [list=a] [*] $g-1$ is also a primitive root $\pmod{p}$. [*] if $p=4k+3$ then $(g-1)^{2k+3} \equiv g-2 \pmod{p}$, and deduce that $g-2$ is also a primitive root $\pmod{p}$. [/list]

Let $z\neq0$ be a complex number satisfying $z^2=z+i|z|$. ($|z|$ denotes the length between the origin and $z$ in the complex plane.) Find $z\cdot\overline{z}$, where $\overline{z}=a-bi$ is the complex conjugate of $z=a+bi$.

Let $P_i$ $i=1,2,......n$ be $n$ points on the plane , $M$ is a point on segment $AB$ in the same plane , prove : $\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| )$. (Here $|AB|$ means the length of segment $AB$) .

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

There are $n>2$ lines given in the plane. No two of the lines are parallel and no three of them intersect at one point. Every point of intersection of these lines is labelled with a natural number between $1$ and $n-1$. Prove that, if and only if $n$ is even, it is possible to assign the labels in such a way that every line has all the numbers from $1$ to $n-1$ at its points of intersection with the other $n-1$ lines.

The rows and columns of a $7\times 7$ grid are each numbered $1,2,\ldots, 7$. In how many ways can one choose 8 cells of this grid such that for every two chosen cells $X$ and $Y$, either the positive difference of their row numbers is at least $3$, or the positive difference of their column numbers is at least $3$? [hide="Clarifications"] [list] [*] The ``or'' here is inclusive (as by convention, despite the ``either''), i.e. $X$ and $Y$ are permitted if and only if they satisfy the row condition, the column condition, or both.[/list][/hide] [i]Ray Li[/i]

For a positive integer $k$, denote by $f(k)$ the number of positive integer $m$ such that the remainder of $km$ modulo $2019^3$ is greater than $m$. Find the amount of different numbers among $f(1), f(2), ..., f(2019^3)$.

Circle $\omega$ lies inside the circle $\Omega$ and touches it internally at point $P$. Point $S$ is taken on $\omega$ and the tangent to $\omega$ is drawn through it. This tangent meets $\Omega$ at points $A$ and $B$. Let $I$ be the centre of $\omega$. Find the locus of circumcentres of triangles $AIB$.

Let $ABC$ be a triangle with $AB > AC$. Let $D$ be a point on the side $AB$ such that $DB = DC$ and let $M$ be the midpoint of $AC$. The line parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC.$