There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?

For a real number $t$ and positive real numbers $a,b$ we have \[2a^2-3abt+b^2=2a^2+abt-b^2=0\] Find $t.$

Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.

On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets. If the expected number of times he writes the number $1$ can be written in the form $\tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i] Proposed by David Altizio [/i]

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$. [i]Proposed by Vincent Huang and Nathan Weckwerth

One of Landau’s four unsolved problems asks whether there are infinitely many primes $p$ such that $p- 1$ is a perfect square. How many such primes are there less than $100$?

Given that $\frac{x}{\sqrt{x} + \sqrt{y}} = 18$ and $\frac{y}{\sqrt{x} + \sqrt{y}} = 2$, find $\sqrt{x} - \sqrt{y}$.

Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.

Let $c \geq 1$ be an integer, and define the sequence $a_1,\ a_2,\ a_3,\ \dots$ by \[ \begin{aligned} a_1 & = 2, \\ a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned} \] Prove that $a_n$ is an integer for all $n$.

Prove that we can draw a line (by a ruler and a compass) from a vertice of a convex quadrilateral such that, the line divides the quadrilateral to two equal areas.

Layna wants to paint a rectangular wall green, but she only has blue and yellow paint. She finds that a $2:1$ mix of blue paint to yellow paint produces the color green she wants, and she knows that one gallon of paint will cover $80$ square feet of wall. If the wall is $8$ feet tall and $21$ feet long, how many gallons of blue paint does Layna need? Express your answer as a fraction in simplest form.

Let $ABC$ be an isosceles triangle, $AB=AC$ inscribed in a circle $\omega$. The $B$-symmedian intersects $\omega$ again at $D$. The circle through $C,D$ and tangent to $BC$ and the circle through $A,D$ and tangent to $CD$ intersect at points $D,X$. The incenter of $ABC$ is denoted $I$. Prove that $B,C,I,X$ are concyclic.

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite. [i]Proposed by A. Golovanov[/i]

A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.

There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?

For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$. Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$. (Richard Henner)

Show that the right circular cylinder of volume $V$ which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)

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 $YX/XZ=BP/PC$.