Five players $(A,B,C,D,E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per a day. What is the least number of days needed for this tournament?

The perpendicular bisector of the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

A [i]uniform covering[/i] of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets). Given two uniform coverings, we define a new uniform covering, their [i]sum[/i] (denoted by $\oplus$), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called [i]non-composite[/i] if it's not a sum of two uniform coverings. Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.

Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$. (Edited)

Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? $\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Let $n$ and $k$ be positive integers with $1 \leq k \leq n$. A set of cards numbered $1$ to $n$ are arranged randomly in a row from left to right. A person alternates between performing the following moves: [list=a] [*] The leftmost card in the row is moved $k-1$ positions to the right while the cards in positions $2$ through $k$ are each moved one place to the left. [*] The rightmost card in the row is moved $k-1$ positions to the left while the cards in positions $n-k+1$ through $n-1$ are each moved one place to the right. [/list] Determine the probability that after some number of moves the cards end up in order from $1$ to $n$, left to right.

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

Find all real solutions of the system: $ x^2\plus{}2yz\equal{}x,$ $ y^2\plus{}2zx\equal{}y,$ $ z^2\plus{}2xy\equal{}z.$

Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? $\textbf{(A) } 11 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 23 \qquad\textbf{(E) } 27$

Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$, where $|z_0|\le 1$.

Bread draws a circle. He then selects four random distinct points on the circumference of the circle to form a convex quadrilateral. Kwu comes by and randomly chooses another 3 distinct points (none of which are the same as Bread's four points) on the circle to form a triangle. Find the probability that Kwu's triangle does not intersect Bread's quadrilateral, where two polygons intersect if they have at least one pair of sides intersecting. [i]Proposed by Nathan Cho[/i]

We choose a point on each side of a parallelogram $ABCD$, let these four points be $P, Q, R$ and $S$. Then we divide the parallelogram into several regions using line segments as shown in the figure. The areas of the grey regions are given, except for one (see the figure). Find the area of the region marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/4/7/dbd009042dabdb2eafc8fc74960e9011038dae.png[/img]

Given a triangle $ABC$ and a point $P$ on the side $AB$ . Let $Q$ be the intersection of the straight line $CP$ (different from $C$) with the circumcicle of the triangle. Prove the inequality $$\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2$$ and that equality holds if and only if the $CP$ is bisector of the angle $ACB$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png[/img]

On a circle $\Gamma$, points $A, B, N, C, D, M$ are chosen in a clockwise order in such a way that $N$ and $M$ are the midpoints of clockwise arcs $BC$ and $AD$ respectively. Let $P$ be the intersection of $AC$ and $BD$, and let $Q$ be a point on line $MB$ such that $PQ$ is perpendicular to $MN$. Point $R$ is chosen on segment $MC$ such that $QB = RC$, prove that the midpoint of $QR$ lies on $AC$.

Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.

The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ : $[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$. Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?

Noah has to fit 8 species of animals into 4 cages of the Arc. He planes to put two species of animal in each cage. It turns out that, for each species of animal, there are at most 3 other species with which it cannot share a cage. Prove that there is a way to assign the animals to the cages so that each species shares a cage with a compatible species.

For any positive integer $n$, let $p(n)$ be the product of its digits in base-$10$ representation. Find the maximum possible value of $\frac{p(n)}{n}$ over all integers $n \ge 10$.

Find all ordered pairs of positive integers $\left(a,b\right)$ such that $$\frac{1}{a}+\frac{a}{b}+\frac{1}{ab}=1.$$

$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.

Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$, and $AB$ of the triangle $ABC$, respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$, and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$.

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$