Find the minimum possible value of |m -n|, where $m$ and $n$ are integers satisfying $m + n = mn - 2021$.

Hen Hao randomly selects two distinct squares on a standard $8\times 8$ chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form $\frac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by James Lin

Let $ABC$ be a triangle with, $AB$ ≠ $AC$, and let $K$ is your incenter. The points $P$ and $Q$ are the points of the intersections of the circumcicle($BCK$) with the line(s) $AB$ and $AC$, respectively. Let $D$ be intersection of $AK$ and $BC$. Show that $P, Q, D$ are collinears.

A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-2.687,0),E); label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$

The system $\begin{cases} x^2 - y^2 = 0 \\ (x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?

Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$ [i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$

In the triangle $ABC,\angle A=\frac{\pi}{3},D,M$ are points on the line $AC$ and $E,N$ are points on the line $AB$ such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$ respectively. Let $L$ be the midpoint of $MN$. Prove that $\angle EDL=\angle ELD$

Given a figure $B$ in the plane, consider the figures $A$, containing $\mathcal B$, with the property [i]$(P)$: a composition of an odd number of central symmetries with centers in $A$ is also a central symmetry with center in $A$.[/i] The task of this problem is to determine the smallest such figure, denoted by $\mathcal A$, that is contained in every figure $A$. (a) Determine the figure $\mathcal A$ if $B$ consists of: $(1)$ two distinct points $I,J$; $(2)$ three non-collinear points $I,J,K$. (b) Determine $\mathcal A$ if $B$ is a circle (with nonzero radius). (c) Give some examples of figures $B$ whose associated figures $\mathcal A$ are mutually distinct and distinct from the above ones.

Consider the positive integers written in the decimal system with $n$ digits, the start of which is not zero and where there are no two sevens next to each other. The number of these numbers is called $u_n$. Derive a relation that expresses $u_{n+2}$ in terms of $u_{n+1}$ and $u_n$.

[u]Round 1[/u] [b]p1.[/b] Ravi has a bag with $100$ slips of paper in it. Each slip has one of the numbers $3, 5$, or $7$ written on it. Given that half of the slips have the number $3$ written on them, and the average of the values on all the slips is $4.4$, how many slips have $7$ written on them? [b]p2.[/b] In triangle $ABC$, point $D$ lies on side $AB$ such that $AB \perp CD$. It is given that $\frac{CD}{BD}=\frac12$, $AC = 29$, and $AD = 20$. Find the area of triangle $BCD$. [b]p3.[/b] Compute $(123 + 4)(123 + 5) - 123\cdot 132$. [u]Round 2[/u] [b]p4. [/b] David is evaluating the terms in the sequence $a_n = (n + 1)^3 - n^3$ for $n = 1, 2, 3,....$ (that is, $a_1 = 2^3 - 1^3$ , $a_2 = 3^3 - 2^3$, $a_3 = 4^3 - 3^3$, and so on). Find the first composite number in the sequence. (An positive integer is composite if it has a divisor other than 1 and itself.) [b]p5.[/b] Find the sum of all positive integers strictly less than $100$ that are not divisible by $3$. [b]p6.[/b] In how many ways can Alex draw the diagram below without lifting his pencil or retracing a line? (Two drawings are different if the order in which he draws the edges is different, or the direction in which he draws an edge is different). [img]https://cdn.artofproblemsolving.com/attachments/9/6/9d29c23b3ca64e787e717ceff22d45851ae503.png[/img] [u]Round 3[/u] [b]p7.[/b] Fresh Mann is a $9$th grader at Euclid High School. Fresh Mann thinks that the word vertices is the plural of the word vertice. Indeed, vertices is the plural of the word vertex. Using all the letters in the word vertice, he can make $m$ $7$-letter sequences. Using all the letters in the word vertex, he can make $n$ $6$-letter sequences. Find $m - n$. [b]p8.[/b] Fresh Mann is given the following expression in his Algebra $1$ class: $101 - 102 = 1$. Fresh Mann is allowed to move some of the digits in this (incorrect) equation to make it into a correct equation. What is the minimal number of digits Fresh Mann needs to move? [b]p9.[/b] Fresh Mann said, “The function $f(x) = ax^2+bx+c$ passes through $6$ points. Their $x$-coordinates are consecutive positive integers, and their y-coordinates are $34$, $55$, $84$, $119$, $160$, and $207$, respectively.” Sophy Moore replied, “You’ve made an error in your list,” and replaced one of Fresh Mann’s numbers with the correct y-coordinate. Find the corrected value. [u]Round 4[/u] [b]p10.[/b] An assassin is trying to find his target’s hotel room number, which is a three-digit positive integer. He knows the following clues about the number: (a) The sum of any two digits of the number is divisible by the remaining digit. (b) The number is divisible by $3$, but if the first digit is removed, the remaining two-digit number is not. (c) The middle digit is the only digit that is a perfect square. Given these clues, what is a possible value for the room number? [b]p11.[/b] Find a positive real number $r$ that satisfies $$\frac{4 + r^3}{9 + r^6}=\frac{1}{5 - r^3}- \frac{1}{9 + r^6}.$$ [b]p12.[/b] Find the largest integer $n$ such that there exist integers $x$ and $y$ between $1$ and $20$ inclusive with $$\left|\frac{21}{19} -\frac{x}{y} \right|<\frac{1}{n}.$$ PS. You had better use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $100$ circles of radius one in the plane. A triangle formed by the centres of any three given circles has area at most $2017$. Prove that there is a line intersecting at least three of the circles.

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$. [i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. [i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then \[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\] where $r_1$ is the exradius corresponding to the vertex $A.$

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

Let $n>1$ be any integer. Define $f,g$ as functions from $\{0,1,2,\cdots,n-1 \}$ to $\{0,1,2,\cdots,n-1\}$ defined as \begin{align*} &f(i)=2i \pmod{n} \\ &g(i)=2i+1 \pmod{n} \end{align*} Show that for any integers $\ell,m \in \{0,1,2,\cdots,n-1 \}$ , there are infinitely many compositions of $f,g$ that map $\ell$ to $m$

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$, then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each. What is the least value of $k_1 + k_2 + k_3$ such that $B$ has a strategy to correctly guess the value of $n$ no matter what $A$ chooses?

Let $p_n$ denote the $n^{\text{th}}$ prime number and define $a_n=\lfloor p_n\nu\rfloor$ for all positive integers $n$ where $\nu$ is a positive irrational number. Is it possible that there exist only finitely many $k$ such that $\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all $i=1,2,\ldots,2020?$ [i]Proposed by Superguy and ayan.nmath[/i]

Let $ABC$ be an acute scalene triangle with incenter $I$, and let $M$ be the circumcenter of triangle $BIC$. Points $D$, $B'$, and $C'$ lie on side $BC$ so that $ \angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ} $. Define $P = \overline{AB} \cap \overline{MC'}$, $Q = \overline{AC} \cap \overline{MB'}$, $S = \overline{MD} \cap \overline{PQ}$, and $K = \overline{SI} \cap \overline{DF}$, where segment $EF$ is a diameter of the incircle selected so that $S$ lies in the interior of segment $AE$. It is known that $KI=15x$, $SI=20x+15$, $BC=20x^{5/2}$, and $DI=20x^{3/2}$, where $x = \tfrac ab(n+\sqrt p)$ for some positive integers $a$, $b$, $n$, $p$, with $p$ prime and $\gcd(a,b)=1$. Compute $a+b+n+p$. [i]Proposed by Evan Chen[/i]

Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$

Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ S$ be the set of all triangles $ ABC$ which have property: $ \tan A,\tan B,\tan C$ are positive integers. Prove that all triangles in $ S$ are similar.

For $k=1,2,\dots$, let $f_k$ be the number of times \[\sin\left(\frac{k\pi x}{2}\right)\] attains its maximum value on the interval $x\in[0,1]$. Compute \[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.