In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $. Find the value of the smallest angle of triangle $ ABC $. (Black Maxim)

Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.

Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is [i]reachable[/i] from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef". Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.

$k$ is a positive integer, $R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}$ for $n=2k$ $R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}$ for $n=2k+1$. A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from $R_{n}$ with the property that any two connected marbles have different numbers on them, we call it [i]nice labeling[/i]. If each one of the marbles are written on some numbers from $R_{n}$ with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to $0$, we call it [i]precise labeling[/i]. $n\geq{3}$, if every mechanism that is labeled [i]nicely[/i] with $R_{n}$, could be labeled [i]precisely[/i] with $R_{m}$, what is the minimal value of $m$?

Let $n\geqslant 3$ be a positiv integer. Ana chooses the positive integers $a_1,a_2,\ldots,a_n$ and for any non-empty subset $A\subseteq\{1,2,\ldots,n\}$ she computes the sum \[s_A=\sum_{k \in A}a_k.\]She orders these sums $s_1\leqslant s_2\leqslant\cdots\leqslant s_{2^n-1}.$ Prove that there exists a subset $B\subseteq\{1,2,\ldots,2^n-1\}$ with $2^{n-2}+1$ elements such that, regardless of the integers $a_1,a_2,\ldots,a_n$ chosen by Ana, these can be determined by only knowing the sums $s_i$ with $i\in B.$

It is known that there are polyhedrons whose faces are more numbered than the vertices. Find the smallest number of triangular faces that such a polyhedron can have.

Let $f$ be a function from R to R. Suppose we have: (1) $f(0)=0$ (2) For all $x, y \in (-\infty, -1) \cup (1, \infty)$, we have $f(\frac{1}{x})+f(\frac{1}{y})=f(\frac{x+y}{1+xy})$. (3) If $x \in (-1,0)$, then $f(x) > 0$. Prove: $\sum_{n=1}^{+\infty} f(\frac{1}{n^2+7n+11}) > f(\frac12)$ with $n \in N^+$.

Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$? \[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3 \]

On a table are a cone, resting on the base, and six equal spheres tangent to the cone. Besides that, each sphere is tangent to the two adjacent spheres. Knowing that the radius $R$ of the base of the cone is half its height and determine the radius $r$ of the spheres.

Is it possible for the projection of the set of points $(x, y, z)$ with $0 \leq x, y, z \leq 1$ onto some two-dimensional plane to be a simple convex pentagon?

Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).

Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$. Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer. (FYROM)

In the trapezoid $ABCD$ ($AB \parallel DC$) the bases have lengths $a$ and $c$ ($c < a$), while the other sides have lengths $b$ and $d$. The diagonals are of lengths $m$ and $n$. It is known that $m^2 + n^2 = (a + c)^2$. a) Find the angle between the diagonals of the trapezoid. b) Prove that $a + c < b + d$. c) Prove that $ac < bd$.

A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.

$25$ point in a plane and for all $3$ points, we find $2$ points such that this $2$ points' distance less than $1$ $cm$ . Prove that at least $13$ points in a circle of radius $1$ $cm$.

Let $ 2^{1110} \equiv n \bmod{1111} $ with $ 0 \leq n < 1111 $. Compute $ n $.

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$

Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.

Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .

We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.

Let $\mathbb{R}*$ denote the set of nonzero real numbers. Find all functions $f:\mathbb{R}* \rightarrow \mathbb{R}*$ such that $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ for every pair of nonzero real numbers $x$ and $y$ with $x^2+y \neq 0$.