Three triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$ are given such that their centres of gravity (intersection points of their medians) lie on a straight line, but no three of the $9$ vertices of the triangles lie on a straight line. Consider the set of $27$ triangles $A_iB_jC_k$ (where $i$, $j$, $k$ take the values $1$, $2$, $3$ independently). Prove that this set of triangles can be divided into two parts of the same total area. (A. Andjans, Riga)

Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board. [i]Proposed by N. Beluhov[/i]

A number is [i]additive[/i] if it has three digits, all of them are different and the sum of two of the digits is equal to the remaining one. (For example, $123 (1+2=3), 945 (4+5=9)$). Find the sum of all additive numbers.

Positive integers $n$ and $k$ satisfying $n\geq 2k+1$ are known to Alice. There are $n$ cards with numbers from $1$ to $n$, randomly shuffled as a deck, face down. On her turn, she does the following in order: (i) She first flips over the top card of the deck, and puts it face up on the table. (ii) Then, if Alice has not signed any card, she can sign the newest card now. The game ends after $2k+1$ turns, and Alice must have signed on some card. Let $A$ be the number on the signed cards, and $M$ be the $(k+1)^{\textup{st}}$ largest number among all $2k+1$ face-up cards. Alice's score is $|M-A|$, and she wants the score to be as close to zero as possible. For each $(n,k)$, find the smallest integer $d=d(n,k)$ such that Alice has a strategy to guarantee her score no greater than $d$. [i]Proposed by usjl[/i]

[u]Round 1[/u] [b]1.1.[/b] Suppose a certain menu has $3$ sandwiches and $5$ drinks. How many ways are there to pick a meal so that you have exactly a drink and a sandwich? [b]1.2.[/b] If $a + b = 4$ and $a + 3b = 222222$, find $10a + b$. [b]1.3.[/b] Compute $$\left\lfloor \frac{2019 \cdot 2017}{2018} \right\rfloor $$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. [u]Round 2[/u] [b]2.1.[/b] Andrew has $10$ water bottles, each of which can hold at most $10$ cups of water. Three bottles are thirty percent filled, five are twenty-four percent filled, and the rest are empty. What is the average amount of water, in cups, contained in the ten water bottles? [b]2.2.[/b] How many positive integers divide $195$ evenly? [b]2.3.[/b] Square $A$ has side length $\ell$ and area $128$. Square $B$ has side length $\ell/2$. Find the length of the diagonal of Square $B$. [u]Round 3[/u] [b]3.1.[/b] A right triangle with area $96$ is inscribed in a circle. If all the side lengths are positive integers, what is the area of the circle? Express your answer in terms of $\pi$. [b]3.2.[/b] A circular spinner has four regions labeled $3, 5, 6, 10$. The region labeled $3$ is $1/3$ of the spinner, $5$ is $1/6$ of the spinner, $6$ is $1/10$ of the spinner, and the region labeled $10$ is $2/5$ of the spinner. If the spinner is spun once randomly, what is the expected value of the number on which it lands? [b]3.3.[/b] Find the integer k such that $k^3 = 8353070389$ [u]Round 4[/u] [b]4.1.[/b] How many ways are there to arrange the letters in the word [b]zugzwang [/b] such that the two z’s are not consecutive? [b]4.2.[/b] If $O$ is the circumcenter of $\vartriangle ABC$, $AD$ is the altitude from $A$ to $BC$, $\angle CAB = 66^o$ and $\angle ABC = 44^o$, then what is the measure of $\angle OAD$ ? [b]4.3.[/b] If $x > 0$ satisfies $x^3 +\frac{1}{x^3} = 18$, find $x^5 +\frac{1}{x^5}$ [u]Round 5[/u] [b]5.1.[/b] Let $C$ be the answer to Question $3$. Neethen decides to run for school president! To be entered onto the ballot, however, Neethen needs $C + 1$ signatures. Since no one else will support him, Neethen gets the remaining $C$ other signatures through bribery. The situation can be modeled by $k \cdot N = 495$, where $k$ is the number of dollars he gives each person, and $N$ is the number of signatures he will get. How many dollars does Neethen have to bribe each person with to get exactly C signatures? [b]5.2.[/b] Let $A$ be the answer to Question $1$. With $3A - 1$ total votes, Neethen still comes short in the election, losing to Serena by just $1$ vote. Darn! Neethen sneaks into the ballot room, knowing that if he destroys just two ballots that voted for Serena, he will win the election. How many ways can Neethen choose two ballots to destroy? [b]5.3.[/b] Let $B$ be the answer to Question $2$. Oh no! Neethen is caught rigging the election by the principal! For his punishment, Neethen needs to run the perimeter of his school three times. The school is modeled by a square of side length $k$ furlongs, where $k$ is an integer. If Neethen runs $B$ feet in total, what is $k + 1$? (Note: one furlong is $1/8$ of a mile). [u]Round 6[/u] [b]6.1.[/b] Find the unique real positive solution to the equation $x =\sqrt{6 + 2\sqrt6 + 2x}- \sqrt{6 - 2\sqrt6 - 2x} -\sqrt6$. [b]6.2.[/b] Consider triangle ABC with $AB = 13$ and $AC = 14$. Point $D$ lies on $BC$, and the lengths of the perpendiculars from $D$ to $AB$ and $AC$ are both $\frac{56}{9}$. Find the largest possible length of $BD$. [b]6.3.[/b] Let $f(x, y) = \frac{m}{n}$, where $m$ is the smallest positive integer such that $x$ and $y$ divide $m$, and $n$ is the largest positive integer such that $n$ divides both $x$ and $y$. If $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, what is the median of the distinct values that $f(a, b)$ can take, where $a, b \in S$? [u]Round 7[/u] [b]7.1.[/b] The polynomial $y = x^4 - 22x^2 - 48x - 23$ can be written in the form $$y = (x - \sqrt{a} - \sqrt{b} - \sqrt{c})(x - \sqrt{a} +\sqrt{b} +\sqrt{c})(x +\sqrt{a} -\sqrt{b} +\sqrt{c})(x +\sqrt{a} +\sqrt{b} -\sqrt{c})$$ for positive integers $a, b, c$ with $a \le b \le c$. Find $(a + b)\cdot c$. [b]7.2.[/b] Varun is grounded for getting an $F$ in every class. However, because his parents don’t like him, rather than making him stay at home they toss him onto a number line at the number $3$. A wall is placed at $0$ and a door to freedom is placed at $10$. To escape the number line, Varun must reach 10, at which point he walks through the door to freedom. Every $5$ minutes a bell rings, and Varun may walk to a different number, and he may not walk to a different number except when the bell rings. Being an $F$ student, rather than walking straight to the door to freedom, whenever the bell rings Varun just randomly chooses an adjacent integer with equal chance and walks towards it. Whenever he is at $0$ he walks to $ 1$ with a $100$ percent chance. What is the expected number of times Varun will visit $0$ before he escapes through the door to freedom? [b]7.3.[/b] Let $\{a_1, a_2, a_3, a_4, a_5, a_6\}$ be a set of positive integers such that every element divides $36$ under the condition that $a_1 < a_2 <... < a_6$. Find the probability that one of these chosen sets also satisfies the condition that every $a_i| a_j$ if $i|j$. [u]Round 8[/u] [b]8.[/b] How many numbers between $1$ and $100, 000$ can be expressed as the product of at most $3$ distinct primes? Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.

Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. [asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy] Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that - the rectangle is covered without gaps and without overlaps - no part of a hook covers area outside the rectangle.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

The board has a natural number greater than $1$. At each step, Igor writes the number $n +\frac{n}{p}$ instead of the number $n$ on the board , where $p$ is some prime divisor of $n$. Prove that if Igor continues to rewrite the number infinite times, then he will choose infinitely times the number $3$ as a prime divisor of $p$. [hide=original wording]На доске записано какое-то натуральное число, большее 1. На каждом шагу Игорь переписывает имеющееся на доске число n на число n +n/p, где p - это какой-нибудь простой делитель числа n. Доказать, что если Игорь будет продолжать переписывать число бесконечно долго, то он бесконечно много раз выберет в качестве простого делителя p число 3.[/hide]

There is at least one friend pair in a class of students with different names. Students in an ordered list of some of the students write the names of all their friends who are not currently written on the blackboard, in order. If each student on the list wrote at least one name on the board and the name of each student with at least one friend on the blackboard at the end of the process, call this list a $golden$ $ list$. Prove that there exists a $golden$ $ list$ such that number of students in this list is even.

A round table is surrounded by $n\geqslant 2$ people, each assigned one of the integers $0, 1,\ldots , n-1$ such that no two people have the same number. In each round, everyone adds their number to their right neighbour’s number, and their new number becomes the remainder of the sum when divided by $n{}.$ We call an initial configuration of integers [i]glorious[/i] if everyone’s number remains the same after some finite number of rounds, never changing again. [list=a] [*]For which integers $n\geqslant 2$ is every initial configuration glorious? [*]For which integers $n\geqslant 2$ is there no glorious initial configuration at all? [/list]

Several points are given in the plane. Suppose that for any three of them, there exists an orthogonal coordinate system (determined by the two axes and the unit length) in which these three points have integer coordinates. Prove that there exists an orthogonal coordinate system in which all the given points have integer coordinates.

[u]Round 9[/u] [b]p25.[/b] A positive integer is called spicy if it is divisible by the sum if its digits. Find the number of spicy integers between $100$ and $200$ inclusive. [b]p26.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE} = \frac{BF}{FC} =\frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p27.[/b] Find the largest value of $n$ for which $3^n$ divides ${100 \choose 33}$. [u]Round 10[/u] [b]p28.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle such that $AB \parallel CD$, $AB = 2$, $CD = 4$, and $AC = 9$. What is the radius of the circle? [b]p29.[/b] Find the product of all possible positive integers $n$ less than $11$ such that in a group of $n$ people, it is possible for every person to be friends with exactly $3$ other people within the group. Assume that friendship is amutual relationship. [b]p30.[/b] Compute the infinite product $$\left( 1+ \frac{1}{2^1} \right) \left( 1+ \frac{1}{2^2} \right) \left( 1+ \frac{1}{2^4} \right) \left( 1+ \frac{1}{2^8} \right) \left( 1+ \frac{1}{2^{16}} \right) ...$$ [u]Round 11[/u] [b]p31.[/b] Find the sum of all possible values of $x y$ if $x +\frac{1}{y}= 12$ and $\frac{1}{x}+ y = 8$. [b]p32.[/b] Find the number of ordered pairs $(a,b)$, where $0 < a,b < 1999$, that satisfy $a^2 +b^2 \equiv ab$ (mod $1999$) [b]p33.[/b] Let $f :N\to Q$ be a function such that $f(1) =0$, $f (2) = 1$ and $f (n) = \frac{f(n-1)+f (n-2)}{2}$ . Evaluate $$\lim_{n\to \infty} f (n).$$ [u]Round 12[/u] [b]p34.[/b] Estimate the sumof the digits of $2018^{2018}$. The number of points you will receive is calculated using the formula $\max \,(0,15-\log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p35.[/b] Let $C(m,n)$ denote the number of ways to tile an $m$ by $n$ rectangle with $1\times 2$ tiles. Estimate $\log_{10}(C(100, 2))$. The number of points you will recieve is calculated using the formula $\max \,(0,15- \log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p36.[/b] Estimate $\log_2 {1000 \choose 500}$. The number of points you earn is equal to $\max \,(0,15-|A-E|)$, where $A$ is the true value and $E$ is your estimate. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S_n$ be the number of sequences $(a_1, a_2, \ldots, a_n),$ where $a_i \in \{0,1\},$ in which no six consecutive blocks are equal. Prove that $S_n \rightarrow \infty$ when $n \rightarrow \infty.$

Sasha wants to bake $6$ cookies in his $8$ inch $\times$ $8$ inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly $3$ inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)

Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

Let $k$ be a positive integer. Arnaldo and Bernaldo play a game in a table $2020\times 2020$, initially all the cells are empty. In each round a player chooses a empty cell and put one red token or one blue token, Arnaldo wins if in some moment, there are $k$ consecutive cells in the same row or column with tokens of same color, if all the cells have a token and there aren't $k$ consecutive cells(row or column) with same color, then Bernaldo wins. If the players play alternately and Arnaldo goes first, determine for which values of $k$, Arnaldo has the winning strategy.

We are given a regular decagon with all diagonals drawn. The number "$+ 1$ " is attached to each vertex and to each point where diagonals intersect (we consider only internal points of intersection). We can decide at any time to simultaneously change the sign of all such numbers along a given side or a given diagonal . Is it possible after a certain number of such operations to have changed all the signs to negative?

Five of James’ friends are sitting around a circular table to play a game of Fish. James chooses a place between two of his friends to pull up a chair and sit. Then, the six friends divide themselves into two disjoint teams, with each team consisting of three consecutive players at the table. If the order in which the three members of a team sit does not matter, how many possible (unordered) pairs of teams are possible?

Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing $x$ and $y$ stones respectively. In each play, it is possible to perform one of the following operations: 1. Choose two non-empty piles and take one stone from each pile. 2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones. The player who cannot perform either of operations 1 and 2 loses. Determine who has the winning strategy based on $x$ and $y$.

a) Prove that one cannot assign to each vertex of a cube $ 8$ distinct numbers from the set $\{0, 1, 2, 3, . . . , 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is even. b) Prove that one can assign to each vertex of a cube $8$ distinct numbers from the set $\{0, 1, 2, 3, . . . , 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by $3$.

Fix integers $m \geq 3$ and $n \geq 3$. Each cell of an array with $m$ rows and $n$ columns is coloured one of two colours such that: [b](1)[/b] Both colours occur on every column; and [b](2)[/b] On every two rows the cells on the same column share colour on exactly $k$ columns. Show that, if $m$ is odd, then \[\frac{n(m-1)}{2m}\leq k\leq \frac{n(m-2)}{m}\] [i]The Problem Selection Committee[/i]

We have a $9$ by $9$ chessboard with $9$ kings (which can move to any of $8$ adjacent squares) in the bottom row. What is the minimum number of moves, if two pieces cannot occupy the same square at the same time, to move all the kings into an $X$ shape (a $5\times5$ region where there are $5$ kings along each diagonal of the $X$, as shown below)? \begin{tabular}{ c c c c c } O & & & & O \\ & O & & O & \\ & & O & & \\ & O & & O & \\ O & & & & O \\ \end{tabular} [i]Proposed by David Tang[/i]

At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?