Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

Let be a finite set $ S. $ Determine the number of functions $ f:S\rightarrow S $ that satisfy $ f\circ f=f. $

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that: $i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$. $ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$. a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist. b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.

Consider a curve with the following property: [i]inside the curve one can move an inscribed equilateral triangle so that each vertex of the triangle moves along the curve and draws the whole curve[/i]. Clearly, every circle possesses the property. Find a closed planar curve without self-intersections, that has the property but is not a circle.

[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$? [b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$. [b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$. [b]p4.[/b] Circles of radius $7$, $7$, $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices? [b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square. [b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. [b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. [b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers. [b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n\geq6$ be a even natural number. Prove that any cube can be divided in $ \dfrac{3n(n\minus{}2)}4\plus{}2$ cubes.

[b]p1.[/b] The coins in Merryland all have different integer values: there is a single $1$ cent coin, a single $2$ cent coin, etc. What is the largest number of coins that a resident of Merryland can have if we know that their total value does not exceed $2021$ cents? [b]p2.[/b] For every positive integer $k$ let $$a_k = \left(\sqrt{\frac{k + 1}{k}}+\frac{\sqrt{k+1}}{k}-\frac{1}{k}-\sqrt{\frac{1}{k}}\right).$$ Evaluate the product $a_4a_5...a_{99}$. Your answer must be as simple as possible. [b]p3.[/b] Prove that for every positive integer $n$ there is a permutation $a_1, a_2, . . . , a_n$ of $1, 2, . . . , n$ for which $j + a_j$ is a power of $2$ for every $j = 1, 2, . . . , n$. [b]p4.[/b] Each point of the $3$-dimensional space is colored one of five different colors: blue, green, orange, red, or yellow, and all five colors are used at least once. Show that there exists a plane somewhere in space which contains four points, no two of which have the same color. [b]p5.[/b] Suppose $a_1 < b_1 < a_2 < b_2 <... < a_n < b_n$ are real numbers. Let $C_n$ be the union of $n$ intervals as below: $$C_n = [a_1, b_1] \cup [a_2, b_2] \cup ... \cup [a_n, b_n].$$ We say $C_n$ is minimal if there is a subset $W$ of real numbers $R$ for which both of the following hold: (a) Every real number $r$ can be written as $r = c + w$ for some $c$ in $C_n$ and some $w$ in $W$, and (b) If $D$ is a subset of $C_n$ for which every real number $r$ can be written as $r = d + w$ for some $d$ in $D$ and some $w$ in $W$, then $D = C_n$. (i) Prove that every interval $C_1 = [a_1, b_1]$ is minimal. (ii) Prove that for every positive integer $n$, the set $C_n$ is minimal PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap. Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones. Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.

Each square of a $100\times 100$ board is painted with one of $100$ different colours, so that each colour is used exactly $100$ times. Show that there exists a row or column of the chessboard in which at least $10$ colours are used.

[b]p1.[/b] There are $32$ balls in a box: $6$ are blue, $8$ are red, $4$ are yellow, and $14$ are brown. If I pull out three balls at once, what is the probability that none of them are brown? [b]p2.[/b] Circles $A$ and $B$ are concentric, and the area of circle $A$ is exactly $20\%$ of the area of circle $B$. The circumference of circle $B$ is $10$. A square is inscribed in circle $A$. What is the area of that square? [b]p3.[/b] If $x^2 +y^2 = 1$ and $x, y \in R$, let $q$ be the largest possible value of $x+y$ and $p$ be the smallest possible value of $x + y$. Compute $pq$. [b]p4.[/b] Yizheng and Jennifer are playing a game of ping-pong. Ping-pong is played in a series of consecutive matches, where the winner of a match is given one point. In the scoring system that Yizheng and Jennifer use, if one person reaches $11$ points before the other person can reach $10$ points, then the person who reached $11$ points wins. If instead the score ends up being tied $10$-to-$10$, then the game will continue indefinitely until one person’s score is two more than the other person’s score, at which point the person with the higher score wins. The probability that Jennifer wins any one match is $70\%$ and the score is currently at $9$-to-$9$. What is the probability that Yizheng wins the game? [b]p5.[/b] The squares on an $8\times 8$ chessboard are numbered left-to-right and then from top-to-bottom (so that the top-left square is $\#1$, the top-right square is $\#8$, and the bottom-right square is $\#64$). $1$ grain of wheat is placed on square $\#1$, $2$ grains on square $\#2$, $4$ grains on square $\#3$, and so on, doubling each time until every square of the chessboard has some number of grains of wheat on it. What fraction of the grains of wheat on the chessboard are on the rightmost column? [b]p6.[/b] Let $f$ be any function that has the following property: For all real numbers $x$ other than $0$ and $1$, $$f \left( 1 - \frac{1}{x} \right) + 2f \left( \frac{1}{1 - x}\right)+ 3f(x) = x^2.$$ Compute $f(2)$. [b]p7.[/b] Find all solutions of: $$(x^2 + 7x + 6)^2 + 7(x^2 + 7x + 6)+ 6 = x.$$ [b]p8.[/b] Let $\vartriangle ABC$ be a triangle where $AB = 25$ and $AC = 29$. $C_1$ is a circle that has $AB$ as a diameter and $C_2$ is a circle that has $BC$ as a diameter. $D$ is a point on $C_1$ so that $BD = 15$ and $CD = 21$. $C_1$ and $C_2$ clearly intersect at $B$; let $E$ be the other point where $C_1$ and $C_2$ intersect. Find all possible values of $ED$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We say that a table with three rows or infinite columns is [i]cool [/i] if it was filled with natural numbers, and also whenever the same number m appears in two or more different places in the table, the numbers that appear in the cells immediately below said places (when they exist) are equal. For example, the following table is cool: [img]https://cdn.artofproblemsolving.com/attachments/5/7/16583a6a9434fd2792a4df48a733226cf2f560.png[/img] For each of the following two tables, decide whether it is possible to fill in the empty cells before the resulting tables are cool, explaining how to do this, or why it is not possible to do this. In both tables from the fifth column, the number in the third line is two units greater than the number in the first line. [img]https://cdn.artofproblemsolving.com/attachments/8/a/56d2f05ea09555c39da88f09eb5901a57567f0.png[/img]

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

The numbers $1, 2,. . . , 10000, $ were placed in the cells of a $100 \times 100$ square, each once; in this case, numbers differing by $1$ are written in cells adjacent to each side. After that we calculated distances between the centers of every two cells whose numbers differ by exactly $5000$. Let $S$ be the minimum of these distances What is the largest value $S$ can take?

Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$.

We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected. Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.

The bell number $b_n$ is the number of ways to partition the set $\{1,2,\ldots,n\}$. For example $b_3=5$. Find a recurrence for $b_n$ and show that $b_n=e^{-1}\sum_{k\geq 0} \frac{k^n}{k!}$. Using a combinatorial proof show that the number of ways to partition $\{1,2,\ldots,n\}$, such that now two consecutive numbers are in the same block, is $b_{n-1}$.

A $(0_x, 1_y, 2_z)$-string is an infinite ternary string such that: [list] [*] If there is a $0$ in position $i$ then there is a $1$ in position $i + x$, [*] if there is a $1$ in position $j$ then there is a $2$ in position $j + y$, [*] if there is a $2$ in position $k$ then there is a $0$ in position $k + z$. [/list] For how many ordered triples of positive integers $(x, y, z)$ with $x, y, z \leq 100$ does there exist $(0_x, 1_y, 2_z)$-string?

Is it possible to cut a rectangle $2001\times2003$ into pieces of the form [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNS82L2RjZTZjNzc0M2YxMzM1ZDIzZTY2Zjc2NGJlMWJlMWUwMmU2ZWRlLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNS0xMyBhdCAzLjQ2LjQ2IFBNLnBuZw==[/img] each consisting of three unit squares?

Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length $1, 2, 3, \dots$ (each positive integral length occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is $1\times 2$). [asy] unitsize(0.5 cm); for(int i = 1; i <= 9; ++i) { draw((0,i)--(10,i)); } for(int i = 0; i <= 4; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 1,2*j)--(2*i + 1,2*j + 1)); } } for(int i = 0; i <= 3; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2)); } } [/asy]

Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?