We are given the line $ d, $ and a point $ A $ which is not on $ d. $ Two points $ B $ and $ C $ move on $ d $ such that the angle $ \angle BAC $ is constant. Prove that the circumcircle of $ ABC $ is tangent to a fixed circle.

$2019$ coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from $ 1$ to $99$, the second player in one move can take an even number of coins from $2$ to $100$. The player who can not make a move is lost. Who has the winning strategy in this game?

Call a positive integer [I]nice[/I] if the sum of its even proper divisors is larger than the sum of its odd proper divisors. What is the smallest nice number that is congruent to $2 \text{ mod } 4$?

Let $ ABCD$ be a chordal/cyclic quadrilateral. Consider points $ P,Q$ on $ AB$ and $ R,S$ on $ CD$ with \[ \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}.\] How to choose $ P,Q,R,S$ such that $ \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD}$ is minimal?

A sequence $a_1, a_2, a_3, \ldots $ is defined as follows: $a_1$ is a positive integer and $$a_{n+1} = \left\lfloor \frac{3}{2} a_n \right\rfloor +1$$ for all $n \in \mathbb{N}$. Can $a_1$ be chosen in such a way that the first $100000$ terms of the sequence are even, but the $100001$-th term is odd?

Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to \mathbb N$ such that for every $m,n\in \mathbb N$, \[ f(m)+f(n)\mid m+n. \]

$a_1 \le a_2 \le ... \le a_n$ is a sequence of reals $b_1, _b2, b_3,..., b_n$ is any rearrangement of the sequence $B_1 \le B_2 \le ...\le B_n$. Show that $\sum a_ib_i \le \sum a_i B_i$.

Points $A'$ and $B'$ lie inside the parallelogram $ABCD$ and points $C'$ and $D'$ lie outside of it, so that all sides of 8-gon $AA'BB'CC'DD'$ are equal. Prove that $A'$, $B'$, $C'$, $D'$ are concyclic.

In triangle $ABC$, $\angle{BAC}=15^{\circ}$. The circumcenter $O$ of triangle $ABC$ lies in its interior. Find $\angle{OBC}$. [asy] size(3cm,0); dot((0,0)); draw(Circle((0,0),1)); draw(dir(70)--dir(220)); draw(dir(220)--dir(310)); draw(dir(310)--dir(70)); draw((0,0)--dir(220)); label("$A$",dir(70),NE); label("$B$",dir(220),SW); label("$C$",dir(310),SE); label("$O$",(0,0),NE); [/asy] $\text{(A) }30^{\circ}\qquad\text{(B) }75^{\circ}\qquad\text{(C) }45^{\circ}\qquad\text{(D) }60^{\circ}\qquad\text{(E) }15^{\circ}$

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

Let $a,$ $b,$ $c,$ $d$ be four positive reals such that $d=a+b+c+2\sqrt{ab+bc+ca}.$ Prove that $a=b+c+d-2\sqrt{bc+cd+db}.$ Darij Grinberg

Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?

For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.

Let $AB$ and $CD$ be chords in a circle of center $O$ with $A , B , C , D$ distinct , and with the lines $AB$ and $CD$ meeting at a right angle at point $E$. Let also $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively . If $MN \bot OE$ , prove that $AD \parallel BC$.

Find the number of ordered pairs $(a,b)$ of positive integers less than or equal to $20$ such that \[\gcd(a,b)>1 \quad \text{and} \quad \frac{1}{\gcd(a,b)}+\frac{a+b}{\text{lcm}(a,b)} \geq 1.\] [i]Proposed by Zachary Perry[/i]

Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]

Determine the sum of all distinct real values of $x$ such that $||| \cdots ||x|+x| \cdots |+x|+x|=1$ where there are $2017$ $x$s in the equation.

A polynomial whose roots are all equal to each other is called a [i]unicorn[/i]. Compute the number of distinct ordered triples $(M, P, G)$, where $M$, $P$, $G$ are complex numbers, such that the polynomials $z^3 + M z^2 + Pz + G$ and $z^3 + G z^2 + Pz + M$ are both unicorns.

Find all pairs of integers $(x, y)$ such that the number \[ \frac{x^2 + y^2}{xy} \] is an integer.

Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.

Tangents $ l_1$ and $ l_2$ common to circles $ c_1$ and $ c_2$ intersect at point $ P$, whereby tangent points remain to different sides from $ P$ on both tangent lines. Through some point $ T$, tangents $ p_1$ and $ p_2$ to circle $ c_1$ and tangents $ p_3$ and $ p_4$ to circle $ c_2$ are drawn. The intersection points of $ l_1$ with lines $ p_1, p_2, p_3, p_4$ are $ A_1, B_1, C_1, D_1$, respectively, whereby the order of points on $ l_1$ is: $ A_1, B_1, P, C_1, D_1$. Analogously, the intersection points of $ l_2$ with lines $ p_1, p_2, p_3, p_4$ are $ A_2, B_2, C_2, D_2$, respectively. Prove that if both quadrangles $ A_1A_2D_1D_2$ and $ B_1B_2C_1C_2$ are cyclic then radii of $ c_1$ and $ c_2$ are equal.

Let $f = (f_1, f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\tfrac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that \[\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0\] everywhere. Prove that $f$ is one-to-one.

Let $ABCD$ be a cyclic quadrilateral, and $M$, $N$ be the midpoints of arcs $AB$ and $CD$ respectively. Prove that $MN$ bisects the segment between the incenters of triangles $ABC$ and $ADC$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$