There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the intersection and one of $A$, $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns. [i] Proposed by usjl[/i]

Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.) [img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png [/img] Alexandru Girban

Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.

On the circumcircle of the acute triangle $ABC$, $D$ is the midpoint of $ \stackrel{\frown}{BC}$. Let $X$ be a point on $ \stackrel{\frown}{BD}$, $E$ the midpoint of $ \stackrel{\frown}{AX}$, and let $S$ lie on $ \stackrel{\frown}{AC}$. The lines $SD$ and $BC$ have intersection $R$, and the lines $SE$ and $AX$ have intersection $T$. If $RT \parallel DE$, prove that the incenter of the triangle $ABC$ is on the line $RT.$

[b]p1.[/b] Can a number ending in $1999$ be the square of a natural number? [b]p2.[/b] The Three-Headed Snake Gorynych celebrated his birthday. His heads took turns feasting on birthday cakes and ate two identical cakes in $15$ minutes. It is known that each head ate as much time as it would take the other two to eat the same pie together. In how many minutes would the three heads of the Serpent Gorynych eat one pie together? [b]p3.[/b] Find the sum of the coefficients of the polynomial obtained after opening the brackets and bringing similar terms into the expression: a) $(7x - 6)^4 - 1$ b) $(7x - 6)^{1999}-1$ [b]p4.[/b] The general wants to arrange seven anti-aircraft installations so that among any three of them there are two installations, the distance between which is exactly $10$ kilometers. Help the general solve this problem. [b]p5.[/b] Gulliver, whose height is $999$ millimeters, is building a tower of cubes. The first cube has a height of $1/2$ a lilikilometer, the second - $1/4$ a lilikilometer, the third - $1/8$ a lilikilometer, etc. How many cubes will be in the tower when its height exceeds Gulliver's height. ($1$ lilikilometer is equal to $1000$ lilimeters). [b]p6.[/b] It is known that in any pentagon you can choose three diagonals from which you can form a triangle. Is there a pentagon in which such diagonals can be chosen in a unique way? [b]p7.[/b] It is known that for natural numbers $a$ and $b$ the equality $19a = 99b$ holds. Can $a + b$ be a prime number? [b]p8.[/b] Vitya thought of $5$ integers and told Vanya all their pairwise sums: $$0, 1, 5, 7, 11, 12, 18, 24, 25, 29.$$ Help Vanya guess the numbers he has in mind. [b]p9.[/b] In a $3 \times 3$ square, numbers are arranged so that the sum of the numbers in each row, in each column and on each major diagonal is equal to $0$. It is known that the sum of the squares of the numbers in the top row is $n$. What can be the sum of the squares of the numbers in the bottom line? [b]p10.[/b] $N$ points are marked on a circle. Two players play this game: the first player connects two of these points with a chord, from the end of which the second player draws a chord to one of the remaining points so as not to intersect the already drawn chord. Then the first player makes the same “move” - draws a new chord from the end of the second chord to one of the remaining points so that it does not intersect any of the already drawn ones. The one who cannot make such a “move” loses. Who wins when played correctly? (A chord is a segment whose ends lie on a given circle) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here[/url].

Let $g:\mathbb{N} \to \mathbb{N}$ with $g(n)$ being the product of the digits of $n$. (a) Prove that $g(n) \le n$ for all $n\in \mathbb{N}$ (b) Find all $n\in \mathbb{N}$ for which $n^2-12n+36=g(n)$

In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.

A triangle has two sides of lengths $1984$ and $2016$. Find the maximum possible area of the triangle. [i]Proposed by Nathan Ramesh

What is the sum of distinct real roots of $x^5-2x^2-9x-6$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ -2 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ -17 $

Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between $\tfrac{1}{2}$ and $\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning? $ \textbf{(A) }\frac{1}{2}\qquad \textbf{(B) }\frac{13}{24} \qquad \textbf{(C) }\frac{7}{12} \qquad \textbf{(D) }\frac{5}{8} \qquad \textbf{(E) }\frac{2}{3}\qquad $

A circle is divided into $2n$ equal by $2n$ points. Ali draws $n+1$ arcs, of length $1,2,\ldots,n+1$. Prove that we can find two arcs, such that one of them is inside in the other one.

Given a triangle $ABC$, $A_1$ divides the length of the path $CAB$ into two equal parts, and define $B_1$ and $C_1$ analogously. Let $l_A$, $l_B$, $l_C$ be the lines passing through $A_1$, $B_1$ and $C_1$ and being parallel to the bisectors of $\angle A$, $\angle B$, and $\angle C$. Show that $l_A$, $l_B$, $l_C$ are concurrent.

Let $ABC$ be an equilateral triangle and $n\ge 2$ be an integer. Denote by $\mathcal{A}$ the set of $n-1$ straight lines which are parallel to $BC$ and divide the surface $[ABC]$ into $n$ polygons having the same area and denote by $\mathcal{P}$ the set of $n-1$ straight lines parallel to $BC$ which divide the surface $[ABC]$ into $n$ polygons having the same perimeter. Prove that the intersection $\mathcal{A} \cap \mathcal{P}$ is empty. [i]Laurentiu Panaitopol[/i]

If $0<a_1< \cdots < a_n$, show that the following equation has exactly $n$ roots. $$ \frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+ \frac{a_3}{a_3-x}+ \cdots + \frac {a_n}{a_n - x} = 2015$$

Let $\lambda$ be a positive real number. Inequality $|\lambda xy+yz|\le \dfrac{\sqrt5}{2}$ holds for arbitrary real numbers $x, y, z$ satisfying $x^2+y^2+z^2=1$. Find the maximal value of $\lambda$.

Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$. Evangelos Psychas (Greece)

A train accelerates at $10$ mph/min, and decelerates at $20$ mph/min. The train’s maximum speed is $300$ mph. What’s the shortest amount of the time that the train could take to travel $500$ miles, if it has to be stationary at both the start and end of its trip? Please give your answer in minutes.

Let $t_{k} = a_{1}^k + a_{2}^k +...+a_{n}^k$, where $a_{1}$, $a_{2}$, ... $a_{n}$ are positive real numbers and $k \in \mathbb{N}$. Prove that $$\frac{t_{5}^2 t_1^{6}}{15} - \frac{t_{4}^4 t_{2}^2 t_{1}^2}{6} + \frac{t_{2}^3 t_{4}^5}{10} \geq 0 $$ [i]Proposed by Daniel Velinov[/i]

Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists. [i]Proposed by Carl Schildkraut[/i]

Let $\{x\}$ denote the fractional part of $x$, which means the unique real $0\leq\{x\}<1$ such that $x-\{x\}$ is an integer. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $[m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and $b$ range along all real numbers.

The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$.The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$. Prove that $CF$ is the altitude of triangle $ABC$.

A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

Let $n \ge 3$ be an integer and $S$ be a set of $n$ elements. Determine the largest integer $k_n$ such that: for each selection of $k_n$ $3-$subsets of $S$, there exists a way to color elements of $S$ with two colors such that none of the chosen $3-$subset is monochromatic.

Suppose that $p|(2t^2-1)$ and $p^2|(2st+1)$. Prove that $p^2|(s^2+t^2-1)$