The pentagon $ABCDE$ below is such that the quadrilateral $ABCD$ is a square and $BC=DE$. What is the measure of the angle $\angle AEC$?

Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.

Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?

Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$.

Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

$n$ men and one woman are in the meeting room with $n+1$ chairs, each of them having their own seat. Show that the following two number of cases are equal. (1) Number of cases to choose one man to get out of the room, and make the left $n-1$ men to sit to each other's chair. (2) Number of cases to make $n+1$ people to sit to each other's chair.

[b]2.[/b] Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set $\{ x\in \mathbb{R} : K+x \subset A\}$ is not Borel-measurable (here $K+x = \{y+x : y \in K\}$). ([b]M.16[/b]) [M. Laczkovich]

Given two positive integers $x,y$, we define $z=x\,\oplus\,y$ to be the bitwise XOR sum of $x$ and $y$; that is, $z$ has a $1$ in its binary representation at exactly the place values where $x,y$ have differing binary representations. It is known that $\oplus$ is both associative and commutative. For example, $20 \oplus 18 = 10100_2 \oplus 10010_2 = 110_2 = 6$. Given a set $S=\{a_1, a_2, \dots, a_n\}$ of positive integers, we let $f(S) = a_1 \oplus a_2 \oplus a_3\oplus \dots \oplus a_n$. We also let $g(S)$ be the number of divisors of $f(S)$ which are at most $2018$ but greater than or equal to the largest element in $S$ (if $S$ is empty then let $g(S)=2018$). Compute the number of $1$s in the binary representation of $\displaystyle\sum_{S\subseteq \{1,2,\dots, 2018\}} g(S)$. [i]Proposed by Brandon Wang and Vincent Huang

Prove that: $$\left\lfloor10^{n+3}\cdot\sqrt{\overline{\underbrace{11\ldots1}_{2n\text{ times}}}}\right\rfloor=\overline{\underbrace{33\ldots3}_{2n\text{ times}}166}$$ [i]Proposed by Ovidiu Pop[/i]

There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day. How many different sequences of number plates are possible at the end of the day?

Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that \[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]

The distance between Maykop and Belorechensk is $24$ km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most $6$ kmph) or on a bike (with velocity at most $18$ kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after $2$ hours $40$ minutes. (Only one guy may seat on the bike simultaneously). [i]Folclore[/i]

A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$