Call a quadruple of positive integers $(a, b, c, d)$ fruitful if there are infinitely many integers $m$ such that $\text{gcd} (am + b, cm + d) = 2019$. Find all possible values of $|ad-bc|$ over fruitful quadruples $(a, b, c, d)$.

In each one of the given $2019$ boxes, there are $2019$ stones numbered as $1,2,...,2019$ with total mass of $1$ kilogram. In all situations satisfying these conditions, if one can pick stones from different boxes with different numbers, with total mass of at least 1 kilogram, in $k$ different ways, what is the maximal of $k$?

There are 2017 points in a plane. For each pair of these points we mark the middle of the segment they form when connected. What’s the least number of marked points?

The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.

There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.

Prove that if $A{}$ is an $n\times n$ matrix with complex entries such that $A+A^*=A^2A^*$ then $A=A^*$. (Here, we denote by $M^*$ the conjugate transpose $\overline{M}^t$ of the matrix $M{}$).

Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Triangle $ \triangle ABC$ is given. Points $ D$ i $ E$ are on line $ AB$ such that $ D \minus{} A \minus{} B \minus{} E, AD \equal{} AC$ and $ BE \equal{} BC$. Bisector of internal angles at $ A$ and $ B$ intersect $ BC,AC$ at $ P$ and $ Q$, and circumcircle of $ ABC$ at $ M$ and $ N$. Line which connects $ A$ with center of circumcircle of $ BME$ and line which connects $ B$ and center of circumcircle of $ AND$ intersect at $ X$. Prove that $ CX \perp PQ$.

Let m and n be positive integers. Alice wishes to walk from the point $(0, 0)$ to the point $(m,n)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0,1)$ to the point $(m, n + 1)$ in increments of$ (1, 0)$ and $(0,1)$. Find (with proof) the number of ways for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times).

Let $a$ be a positive integer such that $5^{1994}-1\mid a$. Prove that the expression of $a$ in base $5$ contains at least $1994$ nonzero digits.

Consider the ellipsoid$$\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}=1$$($a$ and $b > 0$) and the ellipse $E$ which is the intersection of the ellipsoid with the plane of equation$$mx + ny + pz = 0$$where the point $P = [m, n, p]$ is a random point from the unit sphere $(m^2 + n^2 + p^2 = 1)$. Consider the random variable $A_E$ the area of the ellipse $E$. If the point $P$ is chosen with uniform distribution with respect to the area on the unit sphere, what is the expectation of $A_E$ ?

Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_{1}, y_{2}, ... , y_{n}$ : $(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.$

The set of values of $m$ for which $mx^2-6mx+5m+1>0$ for all real $x$ is $\textbf{(A)}~m<\frac14$ $\textbf{(B)}~m\ge0$ $\textbf{(C)}~0\le m\le\frac14$ $\textbf{(D)}~0\le m<\frac14$

Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase? $ \textbf{(A) }9\qquad\textbf{(B) }18\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24 $ [asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } }[/asy]

For $n \geq 3$ and $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ given real numbers we have the following instructions: - place out the numbers in some order in a ring; - delete one of the numbers from the ring; - if just two numbers are remaining in the ring: let $S$ be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace Afterwards start again with the step (2). Show that the largest sum $S$ which can result in this way is given by the formula \[S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\ [\frac{k}{2}] - 1\end{pmatrix}a_{k}.\]

Evaluate $\int_{\frac{1-\sqrt{5}}{2}}^{\frac{1+\sqrt{5}}{2}} (2x^2-1)e^{2x}dx.$

Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

Let $\Omega$ be the circumcircle of an isosceles trapezoid $ABCD$, in which $AD$ is parallel to $BC$. Let $X$ be the reflection point of $D$ with respect to $BC$. Point $Q$ is on the arc $BC$ of $\Omega$ that does not contain $A$. Let $P$ be the intersection of $DQ$ and $BC$. A point $E$ satisfies that $EQ$ is parallel to $PX$, and $EQ$ bisects $\angle BEC$. Prove that $EQ$ also bisects $\angle AEP$. [i]Proposed by Li4.[/i]

Prove that there exists a subset $S$ of positive integers such that we can represent each positive integer as difference of two elements of $S$ in exactly one way.

Let $ ABCD$ be a quadrilateral inscribed in a circle of center $ O$. Let M and N be the midpoints of diagonals $ AC$ and $ BD$, respectively and let $ P$ be the intersection point of the diagonals $ AC$ and $ BD$ of the given quadrilateral .It is known that the points $ O,M,Np$ are distinct. Prove that the points $ O,N,A,C$ are concyclic if and only if the points $ O,M,B,D$ are concyclic. [i]Proposer[/i]: [b]Dorian Croitoru[/b]

Let $p,q,r$ be prime numbers such that $2p>q$, $q > 2r$ and $q>p+r$. Prove that $p+q+r\ge 20$.

Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.

Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.

For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.