Find the greatest positive integer $n$ such that $2^n$ divides \[\text{lcm}\left(1^1,2^2,3^3,\ldots,2016^{2016}\right).\] [i]Proposed by Michael Tang[/i]

Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.

Define the distance between two triangles to be the closest distance between two vertices, one from each triangle. Is it possible to draw five triangles in the plane such that for any two of them, their distance equals the sum of their circumradii?

In a class with $n$ students in the span of $k$ days, each day are chosen three to be tested. Each two students can be taken in such triple only once. Prove that for the greatest $k$ satisfying these conditions, the following inequalities are true: $\frac{n(n-3)}{6}\leq k\leq \frac{n(n-1)}{6}$.

The altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at point $H$. Let $F$ be the intersection of the line $AB$ and the line that is parallel to the side BC and goes through the circumcenter of $ABC$. Let $M$ be the midpoint of the segment $AH$. Prove that $\angle CMF = 90^o$

Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\] Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.

An integer is written in each cell of a board of$ N$ rows and $N + 1$ columns. Prove that some columns (possibly none) can be deleted so that in each row the sum of the numbers left uncrossed out is even.

Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that: $ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum $$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots$$ of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A)} \text{ 16} \qquad \textbf{(B)} \text{ 17} \qquad \textbf{(C)} \text{ 19} \qquad \textbf{(D)} \text{ 23} \qquad \textbf{(E)} \text{ 36}$

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?

There is a unique quadruple of positive integers $(a,b,c,k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^4-20x^3+108x^2-kx+9.$ Compute $c.$

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem: Let $k$ be a nonnegative integer. A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex $v$ is called an [i]extended successor[/i] of a vertex $u$ if there is a chain of vertices $u_{0}=u$, $u_{1}$, $u_{2}$, ..., $u_{t-1}$, $u_{t}=v$ with $t>0$ such that the vertex $u_{i+1}$ is a successor of the vertex $u_{i}$ for every integer $i$ with $0\leq i\leq t-1$. A vertex is called [i]dynastic[/i] if it has two successors and each of these successors has at least $k$ extended successors. Prove that if the forest has $n$ vertices, then there are at most $\frac{n}{k+2}$ dynastic vertices.

Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$ imply the equation $xy-12x+15y=0$.

A number $N$ has $2009$ positive factors. What is the maximum number of positive factors that $N^2$ could have?

In $R^n$ is given $n-1$ vectors, the coordinates of each are zero-sum integers. Prove that the $(n-1)$-dimensional volume of an $(n-1)$-dimensional parallelepiped $P$ stretched by these vectors, is the product of an integer and $\sqrt(n)$.

Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party

As shown in figure , it is known that $M$ is the midpoint of side $BC$ of $\vartriangle ABC$. $\odot O$ passes through points $A, C$ and is tangent to $AM$. The extension of the segment $BA$ intersects $\odot O$ at point $D$. The lines $CD$ and $MA$ intersect at the point $P$. Prove that $PO \perp BC$. [img]https://cdn.artofproblemsolving.com/attachments/8/a/da3570ec7eb0833c7a396e22ffac2bd8902186.png[/img]

Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.