We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.

Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar? $\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $

Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$

A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. [img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]

A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$. (V Proizvolov)

There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

Every point on a circle is coloured in blue or yellow such there is at least a point of each color. Prove that for every colouring of the circle there is always an isosceles triangle inscribed inside the circle 1) with all vertexes of the same colour. 2) with vertexes of both colours. For every colouring of the circle is there an equilateral triangle inscribed inside the circle 3) with all vertexes of the same colour? 4) with vertexes of both colours?

Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

How many squares must be painted at least on a $5 \times 5$ board such that in each row, in each column and in each $2 \times 2$ square is there at least one square painted?

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.

Three rods of lengths $1396, 1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

Given an ordered $n$-tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$, we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the “[i]innovated tuple[/i]” of $A$. The number of distinct elements in $B$ is called the “[i]innovated degree[/i]” of $A$. Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$

Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$?

Let $P$ be a convex n-gon in the plane with vertices labeled $V_1,...,V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles the blue and green regions are the same. Suppose $P$ has exactly one balancing point/ Show that the balancing point must be a vertex of $P$