Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length $a$, and the remaining three have length $b$ ($a \le b$). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

An angle is given in a plane. Using only a compass, one must find out $(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure. $(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$). $\textit {Proposed by Viktor Simjanoski and Nikola Velov}$

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

Show that any odd number can be written as the difference between two perfect squares.

Let $x, y$ be numbers in the interval (0,1) such that for some $a>0, a \neq 1$ $$\log _{x} a+\log _{y} a=4 \log _{x y} a$$Prove that $x=y$

[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$. [b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make? [b]p3.[/b] Let $ABCD$ be a rhombus such that $\vartriangle ABD$ and $\vartriangle BCD$ are equilateral triangles. Find the angle measure of $\angle ACD$ in degrees. [b]p4.[/b] Find the units digit of $2019^{2019}$. [b]p5.[/b] Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same. [b]p6.[/b] Kathy rolls two fair dice numbered from $1$ to $6$. At least one of them comes up as a $4$ or $5$. Compute the probability that the sumof the numbers of the two dice is at least $10$. [b]p7.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $20x +19y = 2019$. [b]p8.[/b] Let $p$ be a prime number such that both $2p -1$ and $10p -1$ are prime numbers. Find the sum of all possible values of $p$. [b]p9.[/b] In a square $ABCD$ with side length $10$, let $E$ be the intersection of $AC$ and $BD$. There is a circle inscribed in triangle $ABE$ with radius $r$ and a circle circumscribed around triangle $ABE$ with radius $R$. Compute $R -r$ . [b]p10.[/b] The fraction $\frac{13}{37 \cdot 77}$ can be written as a repeating decimal $0.a_1a_2...a_{n-1}a_n$ with $n$ digits in its shortest repeating decimal representation. Find $a_1 +a_2 +...+a_{n-1}+a_n$. [b]p11.[/b] Let point $E$ be the midpoint of segment $AB$ of length $12$. Linda the ant is sitting at $A$. If there is a circle $O$ of radius $3$ centered at $E$, compute the length of the shortest path Linda can take from $A$ to $B$ if she can’t cross the circumference of $O$. [b]p12.[/b] Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan. [b]p13.[/b] Find the number of subsets of $\{1, 2, 3, 4, 5, 6,7\}$ which contain four consecutive numbers. [b]p14.[/b] Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes. [b]p15.[/b] There are $10$ distinct subway lines in Boston, each of which consists of a path of stations. Using any $9$ lines, any pair of stations are connected. However, among any $8$ lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through? [b]p16.[/b] There exist positive integers $k$ and $3\nmid m$ for which $$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.$$ Find the value $k$. [b]p17.[/b] Geronimo the giraffe is removing pellets from a box without replacement. There are $5$ red pellets, $10$ blue pellets, and $15$ white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed. [b]p18.[/b] Find the remainder when $$70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)$$ is divided by $71$. [b]p19.[/b] Let $A_1A_2...A_{12}$ be the regular dodecagon. Let $X$ be the intersection of $A_1A_2$ and $A_5A_{11}$. Given that $X A_2 \cdot A_1A_2 = 10$, find the area of dodecagon. [b]p20.[/b] Evaluate the following infinite series: $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}$$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.

Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.