The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.

The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.

A triangle has altitudes of length 5 and 7. What is the maximum length of the third altitude?

Given a triangle $ABC$, let $D$ be the point where the incircle of the triangle $ABC$ touches the side $BC$. A circle through the vertices $B$ and $C$ is tangent to the incircle of triangle $ABC$ at the point $E$. Show that the line $DE$ passes through the excentre of triangle $ABC$ corresponding to vertex $A$.

Compute the sum \[S=1+2+3-4-5+6+7+8-9-10+\dots-2010\] where every three consecutive $+$ are followed by two $-$.

Prove that for any integer $n$ the number $n (n^2 + 5)$ is divisible by $6$.

Let $ABCDEFGH$ be the rectangular parallelepiped where $ABCD$ and $EFGH$ are squares and the edges $AE,BF,CG,DH$ are all perpendicular to the squares. Prove that if the $12$ edges of the parallelepiped have integer lengths, the internal diagonal $AG$ and the face diagonal $AF$ cannot both have integer length.

Mrs. Jones is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only has enough orange juice to fill one third of the last glass. What fraction of a glass of orange juice does she need to pour from the 3 full glasses into the last glass so that all glasses have an equal amount of orange juice? $\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

On each OMA lottery ticket there is a $9$-digit number that only uses the digits $1, 2$ and $3$ (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the $9$ figures, then they are of different colors. Bill $122222222$ is red, $222222222$ is green, what color is bill $123123123$?

Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively, \[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] if a) $X_1, \ldots , X_n$ are the midpoints of the corressponding sides, b) $X_1, \ldots , X_n$ are the feet of the corressponding altitudes, c) $X_1, \ldots , X_n$ are arbitrary points on the corressponding lines. Modified version of [url=https://artofproblemsolving.com/community/c6h418634p2361975]IMO 2010 SL G3[/url] (it was question c)

In connection with proof in geometry, indicate which one of the following statements is [i]incorrect[/i]: $ \textbf{(A)}\ \text{Some statements are accepted without being proved.}$ $ \textbf{(B)}\ \text{In some instances there is more than one correct order in proving certain propositions.}$ $ \textbf{(C)}\ \text{Every term used in a proof must have been defined previously.}$ $ \textbf{(D)}\ \text{It is not possible to arrive by correct reasoning at a true conclusion if, in the given, there is an untrue proposition.}$ $ \textbf{(E)}\ \text{Indirect proof can be used whenever there are two or more contrary propositions.}$

Let triangle $ABC$ have side lengths $AB=11$, $BC=7$, and $AC=12$. Let $D$ be a point on $AC$ and $E$ be a point on $AB$ such that $\angle CDE=90^\circ$ and the area of triangle $CDE$ is maximized. Find the area of triangle $CDE$.

For $ \phi: \mathbb{N} \mapsto \mathbb{Z}$ let us define \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{Z}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}.\] Prove that if $ M_{\phi_1} \equal{} M_{\phi_2} \neq \emptyset,$ then $ \phi_1 \equal{} \phi_2.$ Does this property remain true if \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{N}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}?\]

From each vertex of triangle $ABC$ we draw two rays, red and blue, symmetric about the angle bisector of the corresponding angle. The circumcircles of triangles formed by the intersection of rays of the same color. Prove that if the circumcircle of triangle $ABC$ touches one of these circles then it also touches to the other one.

Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$ Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

a) Show that there are five integers $A, B, C, D$, and $E$ such that $2018 = A^5 + B^5 + C^5 + D^5 + E^5$ b) Show that there are no four integers $A, B, C$ and $D$ such that $2018 = A^5 + B^5 + C^5 + D^5$

Let $ ABCD$ be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a [i]tangent quadrilateral[/i], we mean a quadrilateral that has an incircle.) Let the incircle of the quadrilateral $ ABCD$ touch its sides $ AB$, $ BC$, $ CD$, and $ DA$ in the points $ K$, $ L$, $ M$, and $ N$, respectively. The exterior angle bisectors of the angles $ DAB$ and $ ABC$ intersect each other at a point $ K^{\prime}$. The exterior angle bisectors of the angles $ ABC$ and $ BCD$ intersect each other at a point $ L^{\prime}$. The exterior angle bisectors of the angles $ BCD$ and $ CDA$ intersect each other at a point $ M^{\prime}$. The exterior angle bisectors of the angles $ CDA$ and $ DAB$ intersect each other at a point $ N^{\prime}$. Prove that the straight lines $ KK^{\prime}$, $ LL^{\prime}$, $ MM^{\prime}$, and $ NN^{\prime}$ are concurrent.

How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^{x}+3=0$? $\textbf {(A) } 0 \qquad \textbf {(B) } 1 \qquad \textbf {(C) } 2 \qquad \textbf {(D) } 3 \qquad \textbf {(E) } 4$

There is a board with the shape of an equilateral triangle with side $n$ divided into triangular cells with the shape of equilateral triangles with side $ 1$ (the figure below shows the board when $n = 4$). Each and every triangular cell is colored either red or blue. What is the least number of cells that can be colored blue without two red cells sharing one side? [img]https://cdn.artofproblemsolving.com/attachments/0/1/d1f034258966b319dc87297bdb311f134497b5.png[/img]

Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things: 1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa) 2. Change the first bit after the first $1$. Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.

Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$

The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation \[ x^4 + *x^3 + *x^2 + *x + * = 0 \] with a positive real number. Who has a winning strategy if the goals of the players are: a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots? b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?

Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$\int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx$$ converges, and is less or equal than $n^{3/2}\pi.$

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?