Side of an equilatreal triangle has the length $n\in\mathbb{N}.$ Each side is divided in $ n $ equal segments by division points. A line parallel with the third side of the triangle is drawn through the division points of every two sides. Let $c_n$ be the number of all rhombuses with sidelength $1$ inside the initial triangle. Prove that the greatest solution $ n $ of the inequation $c_n<2009$ is a prime number.

Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation \[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\] has at least one real root.

Show that \[ a^n + b^n + c^n \geq ab^{n-1} + bc^{n-1} + ca^{n-1} \] for real $a,b,c \geq 0$ and $n$ a positive integer.

Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct, [b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal. [b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational. [i]Marin Tolosi[/i]

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$? $ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$

For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$. Find all positive integers $n$ divisible by $2019$ such that $n=d_{19}\cdot d_{20}$. [i](I. Gorodnin)[/i]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Let $B$ be an integer greater than 10 such that everyone of its digits belongs to the set $\{1,3,7,9\}$. Show that $B$ has a [b]prime divisor[/b] greater than or equal to 11.

Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

A convex angle $xOy$ and a point $M$ inside it are given in the plane. Prove that there is a unique point $P$ in the plane with the following property: - For any line $l$ through $M$, meeting the rays $x$ and $y$ (or their extensions) at $X$ and $Y$, the angle $XPY$ is not obtuse.

A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$. Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

Let $P$ be a point inside triangle $ABC$, and suppose lines $AP$, $BP$, $CP$ meet the circumcircle again at $T$, $S$, $R$ (here $T \neq A$, $S \neq B$, $R \neq C$). Let $U$ be any point in the interior of $PT$. A line through $U$ parallel to $AB$ meets $CR$ at $W$, and the line through $U$ parallel to $AC$ meets $BS$ again at $V$. Finally, the line through $B$ parallel to $CP$ and the line through $C$ parallel to $BP$ intersect at point $Q$. Given that $RS$ and $VW$ are parallel, prove that $\angle CAP = \angle BAQ$.

A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

[u]Round 1[/u] [b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!) [b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started? [b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips? [u]Round 2[/u] [b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine? [b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved? [b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$. [u]Round 3 [/u] [b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$. [b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that $\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$. $\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over. Find the smallest possible number of boxes of food. [b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$. [u]Round 4 [/u] [b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$. [b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$? [b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra? $ \textbf{(A)}\ 7,600 \qquad \textbf{(B)}\ 11,400 \qquad \textbf{(C)}\ 12,500 \qquad \textbf{(D)}\ 15,200 \qquad \textbf{(E)}\ 22,800$

Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.

The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$: $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$.

The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$

The management of a secret society is made up of $4$ people. To admit new partners they use the following criteria: $\bullet$ Only the $4$ members of the directory vote, being able to do it in $3$ ways: in favor, against or abstaining. $\bullet$ Each aspiring partner must obtain at least $2$ votes in favor and none against. At the last management meeting, $8$ requests for admission were examined. Of the total votes cast, there were $23$ votes in favor, $2$ votes against and $7$ abstaining. What is the highest and what is the lowest value that the number of approved admission requests can have on that occasion?

Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $ [b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication. [b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups? [b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $

If $a, b, c$ are non-negative real numbers with $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 1$, prove that: \[ \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2\]

The golden ratio $\phi = \frac{1+\sqrt5}{2}$ satisfies the property $\phi^2 =\phi + 1$. Point $P$ lies inside equilateral triangle $\vartriangle ABC$ such that $PA = \phi$, $PB = 2$, and angle $\angle APC$ measures $150$ degrees. What is the measure of $\angle BPC$ in degrees?

The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. After the competition, they queue in front of the restaurant according to the following rules. [list] [*]The Jury chooses the initial order of the contestants in the queue. [*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$. [list] [*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions. [*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends. [/list] [/list] [list=a] [*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices. [*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves. [/list]

Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Katie’s score is strictly greater than Allie’s?