Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.

A container is shaped like a right circular cone open at the top surmounted by a frustum which is open at the top and bottom as shown below. The lower cone has a base with radius 2 centimeters and height 6 centimeters while the frustum has bases with radii 2 and 8 centimeters and height 6 centimeters. If there is a rainfall measuring 2 centimeter of rain, the rain falling into the container will fill the container to a height of $m + 3\sqrt{n}$ cm, where m and n are positive integers. Find m + n.

In $\triangle ABC, \angle A = 100^\circ, \angle B = 50^\circ, \angle C = 30^\circ, \overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC =$ [asy] draw((0,0)--(16,0)--(6,6)--cycle); draw((6,6)--(6,0)--(11,3)--(0,0)); dot((6,6)); dot((0,0)); dot((11,3)); dot((6,0)); dot((16,0)); label("A", (6,6), N); label("B", (0,0), W); label("C", (16,0), E); label("H", (6,0), S); label("M", (11,3), NE);[/asy] $\text{(A)} \ 15^\circ \qquad \text{(B)} \ 22.5^\circ \qquad \text{(C)} \ 30^\circ \qquad \text{(D)} \ 40^\circ \qquad \text{(E)} \ 45^\circ$

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Suppose the following three conditions hold: [list] [*]The smallest of a common internal tangent of $\omega_1$ and $\omega_2$ is equal to $19.$ [*]The length of a common external tangent of $\omega_1$ and $\omega_2$ is equal to $37.$ [*]If two points $X$ and $Y$ are selected on $\omega_1$ and $\omega_2,$ respectively, uniformly at random, then the expected value of $XY^2$ is $2023.$ [/list] Compute the distance between the centers of $\omega_1$ and $\omega_2.$

Prove that every integer $n \ge 12$ is the sum of two composite numbers.

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $. Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush: Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Let $ABC$ be a triangle with $AB<AC$ and let $M $ be the midpoint of $BC$. $MI$ ($I$ incenter) intersects $AB$ at $D$ and $CI$ intersects the circumcircle of $ABC$ at $E$. Prove that $\frac{ED }{ EI} = \frac{IB }{IC}$ [img]https://cdn.artofproblemsolving.com/attachments/0/5/4639d82d183247b875128a842a013ed7415fba.jpg[/img] [hide=.][url=http://artofproblemsolving.com/community/c6h602657p10667541]source[/url], translated by Antreas Hatzipolakis in fb, corrected by me in order to be compatible with it's figure[/hide]

Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$.

Say that a positive integer is [i]red[/i] if it is of the form $n^{2020}$, where $n$ is a positive integer. Say that a positive integer is [i]blue[/i] if it is not red and is of the form $n^{2019}$, where $n$ is a positive integer. True or false: Between every two different red positive integers greater than $10^{100{,}000{,}000}$, there are at least 2019 blue positive integers. Prove that your answer is correct.

A polygon is decomposed into triangles by drawing some non-intersecting interior diagonals in such a way that for every pair of triangles of the triangulation sharing a common side, the sum of the angles opposite to this common side is greater than $180^o$. a) Prove that this polygon is convex. b) Prove that the circumcircle of every triangle used in the decomposition contains the entire polygon. [i]Proposed by Morteza Saghafian - Iran[/i]

Positive real numbers are arranged in the form: $ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$ $ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$ $ 4 \ \ \ 8 \ \ \ 13 ...$ $ 7 \ \ \ 12 ...$ $ 11 ...$ Find the number of the line and column where the number 2002 stays.

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that $\text{(i)}$ $B,C,Y,X$ are concyclic. $\text{(ii)}$ $I$ is also the incentre of triangle $DYX$.

Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions: $\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$ $\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and $\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$ (Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)

Positive integers $n$, $k>1$ are given. Pasha and Vova play a game on a board $n\times k$. Pasha begins, and further they alternate the following moves. On each move a player should place a border of length 1 between two adjacent cells. The player loses if after his move there is no way from the bottom left cell to the top right without crossing any order. Determine who of the players has a winning strategy.

Let $x, y$ and $k$ be three positive integers. Prove that there exist a positive integer $N$ and a set of $k + 1$ positive integers $\{b_0,b_1, b_2, ... ,b_k\}$, such that, for every $i = 0, 1, ... , k$ , the $b_i$-ary expansion of $N$ is a $3$-digit palindrome, and the $b_0$-ary expansion is exactly $\overline{\mbox{xyx}}$. proposed by Bojan Basic, Serbia

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying \[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\] [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

Triangle $ ABC$ is scalene with angle $ A$ having a measure greater than 90 degrees. Determine the set of points $ D$ that lie on the extended line $ BC$, for which \[ |AD|\equal{}\sqrt{|BD| \cdot |CD|}\] where $ |BD|$ refers to the (positive) distance between $ B$ and $ D$.

We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible?

On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_n(x_n, y_n)$ in which $(x_n)^{\infty}_{n=1}$,$(y_n)^{\infty}_{n=1}$ are two sequences of positive numbers satisfing the following conditions: $$x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2 \,\, \forall n \ge 1 $$ Suppose that $O, A_1, A_{2016}$ belong to a line $d$ and $A_1, A_{2016}$ are distinct. Prove that all the points $A_2, A_3,. .. , A_{2015}$ lie on one side of $d$.

Find all real numbers $x$ such that $\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$. (For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)

For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.

Alice has an isosceles triangle $M_0N_0P$, where $M_0P=N_0P$ and $\angle M_0PN_0=\alpha^{\circ}$. (The angle is measured in degrees.) Given a triangle $M_iN_jP$ for nonnegative integers $i$ and $j$, Alice may perform one of two [i]elongations[/i]: a) an $M$-[i]elongation[/i], where she extends ray $\overrightarrow{PM_i}$ to a point $M_{i+1}$ where $M_iM_{i+1}=M_iN_j$ and removes the point $M_i$. b) an $N$-[i]elongation[/i], where she extends ray $\overrightarrow{PN_j}$ to a point $N_{j+1}$ where $N_jN_{j+1}=M_iN_j$ and removes the point $N_j$. After a series of $5$ elongations, $k$ of which were $M$-elongations, Alice finds that triangle $M_kN_{5-k}P$ is an isosceles triangle. Given that $10\alpha$ is an integer, compute $10\alpha$. [i]Proposed by Yannick Yao[/i]

Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.

Let $ P$ be a point inside a regular tetrahedron $ T$ of unit volume. The four planes passing through $ P$ and parallel to the faces of $ T$ partition $ T$ into 14 pieces. Let $ f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $ f(P)$ as $ P$ varies over $ T.$