In triangle ABC, $\angle$ A equals 120 degrees. A point D is inside the triangle such that $\angle$DBC = 2 $\times \angle $ABD and $\angle$DCB = 2 $\times \angle$ACD. Determine the measure, in degrees, of $\angle$ BDC. [asy] pair A = (5,4); pair B = (0,0); pair C = (10,0); pair D = (5,2.5) ; draw(A--B); draw(B--C); draw(C--A); draw (B--D--C); label ("A", A, dir(45)); label ("B", B, dir(45)); label ("C", C, dir(45)); label ("D", D, dir(45)); [/asy]

Show that if $x,y,z,w$ are positive reals, then \[ \frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w) \]

Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values. [i]Note:[/i] For any $d>0$, $\lfloor \log_2 d\rfloor$ is the unique integer $k$ such that $2^k\le d<2^{k+1}$. [i]Proposed by Pranjal Srivastava[/i]

Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that \[\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.\] Determine when equality holds.

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$ $\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$

Find all solutions of the system of equations $\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$ where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$ is an integer.

A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? $ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $

Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is $|AC|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ \dfrac 32 \qquad\textbf{(D)}\ \sqrt 2 \qquad\textbf{(E)}\ \sqrt 3 $

A polygon is called [i]good [/i] if it satisfies the following conditions: (i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$, (ii) It is not self-intersecing, (iii) For any three sides, two are parallel and equal. Find all $n$ for which there exists a [i]good [/i] $n$-gon.

Prove that if every side of a triangle is less than $ 1 $, then its area is less than $ \frac{\sqrt{3}}{4} $.

For a positive integer $n$, let $\mathcal{D}(n)$ be the value obtained by, starting from the left, alternating between adding and subtracting the digits of $n$. For example, $\mathcal{D}(321)=3-2+1=2$, while $\mathcal{D}(40)=4-0=4$. Compute the value of the sum \[\sum_{n=1}^{100}\mathcal{D}(n)=\mathcal{D}(1)+\mathcal{D}(2)+\dots+\mathcal{D}(100).\]

In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.

[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $ [b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $ [i]Florian Dumitrel[/i]

There are 111 StarCraft programmers. The StarCraft team SKT starts with a given set of eleven programmers on it, and at the end of each season, it drops a progamer and adds a programmer (possibly the same one). At the start of the second season, SKT has to field a team of five programmers to play the opening match. How many different lineups of ve players could be fielded if the order of players on the lineup matters?

Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove $ AB\plus{}BC\plus{}CA\geq 8DE$

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

$S$ is natural, and $S=d_{1}>d_2>...>d_{1000000}=1$ are all divisors of $S$. What minimal number of divisors can have $d_{250}$?

[b]p1.[/b] One out of $12$ coins is counterfeited. It is known that its weight differs from the weight of a valid coin but it is unknown whether it is lighter or heavier. How to detect the counterfeited coin with the help of four trials using only a two-pan balance without weights? [b]p2.[/b] Below a $3$-digit number $c d e$ is multiplied by a $2$-digit number $a b$ . Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits. $\begin{tabular}{ccccc} & & c & d & e \\ x & & & a & b \\ \hline & & f & e & g \\ + & c & d & e & \\ \hline & b & b & c & g \\ \end{tabular}$ [b]p3.[/b] Find all integer $n$ such that $\frac{n + 1}{2n - 1}$is an integer. [b]p4[/b]. There are several straight lines on the plane which split the plane in several pieces. Is it possible to paint the plane in brown and green such that each piece is painted one color and no pieces having a common side are painted the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$?

The [b]Collaptz function[/b] is defined as $$C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \\ \frac{n}{2} & n\textrm{~even}.\end{cases}$$ We obtain the [b]Collaptz sequence[/b] of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. Find the sum of the three smallest positive integers $n$ whose Collaptz sequences do not contain $1,$ or in other words, do not [b]collaptzse[/b]. [i]Proposed by Andrew Wu and Jason Wang[/i]

Carol places a king on a $5 \times 5$ chessboard. The king starts on the lower-left corner, and each move it steps one square to the right, up, up-right, up-left, or down-right. How many ways are there for the king to get to the top-right corner without visiting the same square twice?