This season, there are $3n + 1$ teams in the MLS (Major League Soccer). As of now, each team has played exactly $n -1$ matches. Prove that there exist $4$ teams such that none of the $4$ teams have faced each other.

A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square. (V Proizvolov)

In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at $(0,0)$ and the line segment from $(-1,0)$ to $(1,0)$. David stands at the point $(-1,0)$ and shines a flashlight into the room at an angle of $46^{\circ}$ above the horizontal. How many times does the light beam reflect off the walls before coming back to David at $(-1,0)$ for the first time?

$n\geq3$ boxes are placed around a circle. At the first step we choose some boxes. At the second step for each chosen box we put a ball into the chosen box and into each of its two neighbouring boxes. Find the total number of possible distinct ball distributions which can be obtained in this way. (All balls are identical.)

The vertices of a regular $45$-gon are painted into three colors so that the number of vertices of each color is the same. Prove that three vertices of each color can be selected so that three triangles formed by the chosen vertices of the same color are all equal.

For $n(n\geq3)$ positive intengers $a_1,a_2,\cdots,a_n$. Put the numbers on a circle. In each operation, calculate difference between two adjacent numbers and take its absolute value. Put the $n$ numbers we get on another ciecle (do not change their order). Find all $n$, satisfying that no matter how $a_1,a_2,\cdots,a_n$ are given, all numbers on the circle are equal after limited operations.

Given a connected graph with $n$ edges, where there are no parallel edges. For any two cycles $C,C'$ in the graph, define its [i]outer cycle[/i] to be \[C*C'=\{x|x\in (C-C')\cup (C'-C)\}.\] (1) Let $r$ be the largest postive integer so that we can choose $r$ cycles $C_1,C_2,\ldots,C_r$ and for all $1\leq k\leq r$ and $1\leq i$, $j_1,j_2,\ldots,j_k\leq r$, we have \[C_i\neq C_{j_1}*C_{j_2}*\cdots*C_{j_k}.\] (Remark: There should have been an extra condition that either $j_1\neq i$ or $k\neq 1$) (2) Let $s$ be the largest positive integer so that we can choose $s$ edges that do not form a cycle. (Remark: A more precise way of saying this is that any nonempty subset of these $s$ edges does not form a cycle) Show that $r+s=n$. Note: A cycle is a set of edges of the form $\{A_iA_{i+1},1\leq i\leq n\}$ where $n\geq 3$, $A_1,A_2,\ldots,A_n$ are distinct vertices, and $A_{n+1}=A_1$.

There's a convex $3n$-polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots $A$, $B$ and $C$ form a triangle if $A$'s laser points at $B$, $B$'s laser points at $C$, and $C$'s laser points at $A$. Find the minimum number of moves that can guarantee $n$ triangles on the plane.

Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.

$268$ numbers are written around a circle. The $17$th number is $3$, the $83$rd is $4$ and the $144$th is $9$. The sum of every $20$ consecutive numbers is $72$. Find the $210$th number.

There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?

Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?

[u]Round 5[/u] [b]p13.[/b] A $2016 \times 2016$ chess board is cut into $k \ge 1$ rectangle(s) with positive integer sidelengths. Let $p$ be the sum of the perimeters of all $k$ rectangles. Additionally, let $m$ and $M$ be the minimum and maximum possible value of $\frac{p}{k}$, respectively. Determine the ordered pair $(m,M)$. [b]p14.[/b] For nonnegative integers $n$, let $f (n)$ be the product of the digits of $n$. Compute $\sum^{1000}_{i=1}f (i )$. [b]p15.[/b] How many ordered pairs of positive integers $(m,n)$ have the property that $mn$ divides $2016$? [u]Round 6[/u] [b]p16.[/b] Let $a,b,c$ be distinct integers such that $a +b +c = 0$. Find the minimum possible positive value of $|a^3 +b^3 +c^3|$. [b]p17.[/b] Find the greatest positive integer $k$ such that $11^k -2^k$ is a perfect square. [b]p18.[/b] Find all ordered triples $(a,b,c)$ with $a \le b \le c$ of nonnegative integers such that $2a +2b +2c = ab +bc +ca$. [u]Round 7[/u] [b]p19.[/b] Let $f :N \to N$ be a function such that $f ( f (n))+ f (n +1) = n +2$ for all positive integers $n$. Find $f (20)+ f (16)$. [b]p20.[/b] Let $\vartriangle ABC$ be a triangle with area $10$ and $BC = 10$. Find the minimum possible value of $AB \cdot AC$. [b]p21.[/b] Let $\vartriangle ABC$ be a triangle with sidelengths $AB = 19$, $BC = 24$, $C A = 23$. Let $D$ be a point on minor arc $BC$ of the circumcircle of $\vartriangle ABC$ such that $DB =DC$. A circle with center $D$ that passes through $B$ and $C$ interests $AC$ again at a point $E \ne C$. Find the length of $AE$. [u]Round 8[/u] [b]p22.[/b] Let $m =\frac12 \sqrt{2+\sqrt{2+... \sqrt2}}$, where there are $2014$ square roots. Let $f_1(x) =2x^2 -1$ and let $f_n(x) = f_1( f_{n-1}(x))$. Find $f_{2015}(m)$. [b]p23.[/b] How many ordered triples of integers $(a,b,c)$ are there such that $0 < c \le b \le a \le 2016$, and $a +b-c = 2016$? [b]p24.[/b] In cyclic quadrilateral $ABCD$, $\angle B AD = 120^o$,$\angle ABC = 150^o$,$CD = 8$ and the area of $ABCD$ is $6\sqrt3$. Find the perimeter of $ABCD$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

[u]Set 1[/u] [b]p1.[/b] Find $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11$. [b]p2.[/b] Find $1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6$. [b]p3.[/b] Let $\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}$ , where $m$ and $n$ are positive integers that share no prime divisors. Find $m + n$. [u]Set 2[/u] [b]p4.[/b] Define $0! = 1$ and let $n! = n \cdot (n - 1)!$ for all positive integers $n$. Find the value of $(2! + 0!)(1! + 8!)$. [b]p5.[/b] Rachel’s favorite number is a positive integer $n$. She gives Justin three clues about it: $\bullet$ $n$ is prime. $\bullet$ $n^2 - 5n + 6 \ne 0$. $\bullet$ $n$ is a divisor of $252$. What is Rachel’s favorite number? [b]p6.[/b] Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday? [u]Set 3[/u] [b]p7.[/b] Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$. [b]p8.[/b] Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at $(0, 0)$. For each move, he is allowed to select a token on any point $(x, y)$ and take it off the plane, replacing it with two tokens, one at $(x + 1, y)$, and one at $(x, y + 1)$. At the end of the game, for a token on $(a, b)$, it is assigned a score $\frac{1}{2^{a+b}}$ . These scores are summed for his total score. Determine the highest total score Sebastian can get in $100$ moves. [b]p9.[/b] Find the number of positive integers $n$ satisfying the following two properties: $\bullet$ $n$ has either four or five digits, where leading zeros are not permitted, $\bullet$ The sum of the digits of $n$ is a multiple of $3$. [u]Set 4[/u] [b]p10.[/b] [i]A unit square rotated $45^o$ about a vertex, Sweeps the area for Farmer Khiem’s pen. If $n$ is the space the pigs can roam, Determine the floor of $100n$.[/i] If $n$ is the area a unit square sweeps out when rotated 4$5$ degrees about a vertex, determine $\lfloor 100n \rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.png[/img] [b]p11.[/b][i] Michael is planting four trees, In a grid, three rows of three, If two trees are close, Then both are bulldozed, So how many ways can it be?[/i] In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side. [b]p12.[/b] [i]Three sixty-seven Are the last three digits of $n$ cubed. What is $n$?[/i] If the last three digits of $n^3$ are $367$ for a positive integer $n$ less than $1000$, determine $n$. [u]Set 5[/u] [b]p13.[/b] Determine $\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}$. [b]p14. [/b]Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$? [b]p15.[/b] Determine the sum of all positive integers $n$ such that $4n^4 + 1$ is prime. [u]Set 6[/u] [b]p16.[/b] Suppose that $p, q, r$ are primes such that $pqr = 11(p + q + r)$ such that $p\ge q \ge r$. Determine the sum of all possible values of $p$. [b]p17.[/b] Let the operation $\oplus$ satisfy $a \oplus b =\frac{1}{1/a+1/b}$ . Suppose $$N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),$$ where there are $2018$ instances of $\oplus$ . If $N$ can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers, then determine $m + n$. [b]p18.[/b] What is the remainder when $\frac{2018^{1001} - 1}{2017}$ is divided by $2017$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777307p24369763]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

Let $n$ be a positive integer and $S = \{ m \mid 2^n \le m < 2^{n+1} \}$. We call a pair of non-negative integers $(a, b)$ [i]fancy[/i] if $a + b$ is in $S$ and is a palindrome in binary. Find the number of [i]fancy[/i] pairs $(a, b)$.

Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$. We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$. Show that $n=23$ if $T$ has a subset $S$ satisfying [list=i] [*]$|S|=2^k$ [*]For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$[/list]

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

Consider finitely many points in the plane such that no three are collinear. Prove that we can paint the points with two colors such that there is no half-plane that contains exactly three points such that those three points have the same color.

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

Let $S_n$ be the set of points $(x/2,y/2)\in\mathbb{R}^2$ such that $x$ and $y$ are odd integers and $|x|\leq y\leq 2n$. Let $T_n$ be the number of graphs $G$ with vertex set in $S_n$ satisfying the following conditions: [list] [*]G has no cycles. [*]If two points share an edge, then the distance between them is $1$. [*]For any path $P = (a,\dots,b)$ in $G$, the smallest $y$-coordinate among the points in $P$ is either that of $a$ or that of $b$. However, multiple points may share this $y$-coordinate. [/list] Find the $100$th-smallest positive integer $n$ such that the units digit of $T_{3n}$ is $4$.

In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.

Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: [i]for every[/i] $k, k\le n$[i], the length of the sum of the first[/i] $k$ [i]vectors is less than[/i] $2$.