In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

A [b][u]word[/u][/b] is formed by a number of letters of the alphabet. We show words with capital letters. A [b][u]sentence[/u][/b] is formed by a number of words. For example if $A=aa$ and $B=ab$ then the sentence $AB$ is equivalent to $aaab$. In this language, $A^n$ indicates $\underbrace{AA \cdots A}_{n}$. We have an equation when two sentences are equal. For example $XYX=YZ^2$ and it means that if we write the alphabetic letters forming the words of each sentence, we get two equivalent sequences of alphabetic letters. An equation is [b][u]simplified[/u][/b], if the words of the left and the right side of the sentences of the both sides of the equation are different. Note that every word contains one alphabetic letter at least. $\text{a})$We have a simplified equation in terms of $X$ and $Y$. Prove that both $X$ and $Y$ can be written in form of a power of a word like $Z$.($Z$ can contain only one alphabetic letter). $\text{b})$ Words $W_1,W_2,\cdots , W_n$ are the answers of a simplified equation. Prove that we can produce these $n$ words with fewer words. $\text{c})$ $n$ words $W_1,W_2,\cdots , W_n$ are the answers of a simplified system of equations. Define graph $G$ with vertices ${1,2 \cdots ,n}$ such that $i$ and $j$ are connected if in one of the equations, $W_i$ and $W_j$ be the two words appearing in the right side of each side of the equation.($\cdots W_i = \cdots W_j$). If we denote by $c$ the number of connected components of $G$, prove that these $n$ words can be produced with at most $c$ words. [i]Proposed by Mostafa Einollah Zadeh Samadi[/i]

In a round-robin tournament with $n$ players $P_1$, $P_2$, $\ldots$, $P_n$ (where $n > 1$), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $w_r$ and $l_r$ be the number of games won and lost, respectively, by $P_r$. Show that \[ \sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2. \]

Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$. (a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle. (b) Prove that the line $ST$ is parallel to the side $BC$. (Karl Czakler)

Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer.

There are $n$ boxes ${B_1},{B_2},\ldots,{B_n}$ from left to right, and there are $n$ balls in these boxes. If there is at least $1$ ball in ${B_1}$, we can move one to ${B_2}$. If there is at least $1$ ball in ${B_n}$, we can move one to ${B_{n - 1}}$. If there are at least $2$ balls in ${B_k}$, $2 \leq k \leq n - 1$ we can move one to ${B_{k - 1}}$, and one to ${B_{k + 1}}$. Prove that, for any arrangement of the $n$ balls, we can achieve that each box has one ball in it.

A sequence with first term $a_0$ is defined such that $a_{n+1}=2a_n^2-1$ for $n\geq0.$ Let $N$ denote the number of possible values of $a_0$ such that $a_0=a_{2020}.$ Find the number of factors of $N.$ [i]Proposed by Alex Li[/i]

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

Determine the number of 10-letter strings consisting of $A$s, $B$s, and $C$s such that there is no $B$ between any two $A$s.

The diagonals of a convex quadrilateral $ABCD$ meet at point $L$. The orthocenter $H$ of the triangle $LAB$ and the circumcenters $O_1, O_2$, and $O_3$ of the triangles $LBC, LCD$, and $LDA$ were marked. Then the whole configuration except for points $H, O_1, O_2$, and $O_3$ was erased. Restore it using a compass and a ruler.

Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\ y^2 - 3z + p = x, \\ z^2 - 3x + p = y \end{cases}$ with real parameter $p$. a) For $p \ge 4$, solve the considered system in the field of real numbers. b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$. (Jaroslav Svrcek)

To the nearest degree, find the measure of the largest angle in a triangle with side lengths $3$, $5$, and $7$.

Along a track for cross-country skiing, $1000$ seats are placed in a row and numbered in order from $1$ to $1000$. By mistake, $n$ tickets were sold, $100 < n < 1000$, each with one of the numbers $1,2,..., 100$ printed on it. Also for each number $1,2,..., 100$ there exists at least one ticket with this number printed on it. Of course, there are tickets that have the same seat numbers. These $n$ spectators arrive one at a time. Each goes to the seat shown on his ticket and occupies it if it is still empty. If not, he just says “Oh” and moves to the seat with the next number. This is repeated until he finds an empty seat and occupies it, saying “Oh” once for each occupied seat passed over but not at any other time. Prove that all the spectators will be seated and that the total number of the exclamations “Oh” that have been made before all the spectators are seated does not depend on the order in which the n spectators arrive, although it does depend on the distribution of numbers on the tickets. (A Shen)

A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ \sqrt{13}\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

A fudgeflake is a planar fractal figure with a $120^{\circ}$ rotational symmetry such that three identical fudgeflakes in the same orientation fit together without gaps to form a larger fudgeflake with its orientation $30^{\circ}$ clockwise of the smaller fudgeflakes' orientation, as shown below. If the distance between the centers of the original three fudgeflakes is $1$, what is the area of one of those three fudgeflakes? Justify your answer. [asy] defaultpen(linewidth(.7)); string s = "00d08c8520612022202288272220065886,00e0708768822888788866683,01006c8765,01206b88227606,01208c858768588616678868160027,017068870228728868822872272220600278886,0190988886872272882228166060,02209486868506,02304b282022852282022828888828228166060066002000668666,02304d8882858688886166702,023050,0230637222282272220786,0240918681,02505f,0260908527222027285886852728522820766,02905b81,03904422888888766686882288888607,03908c58588685876668688228882078688228822220258587685886166702,049053852282000027888666,0490fa852061112222282282000702202220065868,04a0ae868822888888866602768688228888860728228822820228586888766702,04a0de2821666868822888500602,04b0ac5812022882227200070,04c04a8220228822272012882876882288227606,04c0da288282220228886882288227600660020060668,04c0f2,04e0fa88868588616668688228882078688228822220786,05e0ba878605,05e0d287688872006,05e102786720,05f0b88220228822816606066860,06002786882288886888666868886166600668666868887,0600748207,0600ce88616,0600ff86816,0610258222282220228887882288228202285868886668688228858688228822827066,062023223,0620cd88227600,0620fe8527222027686,064074555555555555555555555555555555555552882288888876668688866606060276866686882288878886668668522888868527282822816660222228588688861668678886166600668666868886660602782228,0650c985877,0680b0865282888882822202288878886668678282858768522882822200602788860660606,06a0fa88886070220,06c0b48830,0700fe8527222206702,0790fb867888666868822888788228722722720128288768886668688866600668666027686882288886888606786882288888607,0870b9822202288222022888886058,0880428582288868886668688228828768822882282166606060602,088070816,08903e7870,08a06c877,08b02a82858868886067222006,08b03b867202285272220786,08c06b8528220676,09003572,0900745555555555555555555555555555555555555555555555555555555555555555555555555558207,0920668858888285272220786,09606c,09812e88228166,09a128868827,09b12682202288222766060061,09c0648867,09c094868672020228160,09d04b86787228828868886668688866616006686605820228,09d0e38207,09e0908282822202288888828282220228886888681666868522888888285228202282886685886066886888606606666678868160027,09f0478765,0a10468822220276866606,0a411e828886882288227606060602,0a90428888681288227600,0aa046,0aa091,0b111f8282886788766866852288586882288228216660606,0b2122,0b30438861,0b5042858868886668688228882078688228822760600660,0b805f28868686868686850633,0be09d,0c3127868681020222022882276066660,0c40de86788228728868886668688866606066860678688228886888681666868527285886872272882228166602,0c504b88688810600220222027686,0c60da86816,0c70a8,0c80d98527222027686,0c90b08886888666868822888886058228822820228287688868166768886816606006786667021,0cb046827222025888685272850,0cc11f8702288888605822882281660602,0d00d588886070220,0d60d98527222070070,0d811a86868850607,0d90b8786783,0da0268768822882876888666868886661600668666868886816611,0da107272858868886166786888616660066870202,0db0b586816,0dc02527201288227600,0dd0b48527222027686,0e301e8861,0e501d88888605822882222027686,0e50b0888860702,0eb0b48527222206702,0f401d88868886166702,0f40b1867888666868822888788228822202786,0f50298822882888860652288227600,0fc02d,1010b98765,10206f82222282220228888882828222006588606600660020066660660066868,10207288228888685866686888681636160706768166686882288878886706786882288868822885228166,10303e28216668688228887685886166702,1030b8220258527227,10502b87,10503a288282220065886066006,10602958882078688866683,10904e8527285886882288227600,10a0b127282887685272858122816,1160ad86885,11706327683,1170ab8216,11904f877,11905f87022872812220605886,119097882886888666868886661600658,11b04e858868886166783"; string[] k = split(s,","); for(string str:k) { string a = substr(str,0,3),b=substr(str,3,3),c=substr(str,6); real x=hex(a),y=hex(b); for (int i=0;i<length(c);++i) { int next = hex(substr(c,i,1)); real yI=(int)((next-next%3)/3),xI=next%3; --xI; --yI; draw((x,-y)--(x+xI,-(y+yI))); x+=xI; y+=yI; } }[/asy]

In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$. [i]Proposed by Eugene Chen [/i]

Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$

A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by members of the audience? $\text{(A)} \ 24 \qquad \text{(B)} \ 27 \qquad \text{(C)} \ 30 \qquad \text{(D)} \ 33 \qquad \text{(E)} \ 36$

[u]Round 1[/u] [b]p1.[/b] Ash is running around town catching Pokémon. Each day, he may add $3, 4$, or $5$ Pokémon to his collection, but he can never add the same number of Pokémon on two consecutive days. What is the smallest number of days it could take for him to collect exactly $100$ Pokémon? [b]p2.[/b] Jack and Jill have ten buckets. One bucket can hold up to $1$ gallon of water, another can hold up to $2$ gallons, and so on, with the largest able to hold up to $10$ gallons. The ten buckets are arranged in a line as shown below. Jack and Jill can pour some amount of water into each bucket, but no bucket can have less water than the one to its left. Is it possible that together, the ten buckets can hold 36 gallons of water? [img]https://cdn.artofproblemsolving.com/attachments/f/8/0b6524bebe8fe859fe7b1bc887ac786106fc17.png[/img] [b]p3.[/b] There are $2023$ knights and liars standing in a row. Knights always tell the truth and liars always lie. Each of them says, “the number of liars to the left of me is greater than the number of knights to the right.” How many liars are there? [b]p4.[/b] Camila has a deck of $101$ cards numbered $1, 2, ..., 101$. She starts with $50$ random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all $50$ cards in her hand with her choice of $50$ of the $51$ cards from the table. Show that Camila can make at most 50 exchanges and end up with cards $1, 2, ..., 50$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.png[/img] [b]p5.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [u]Round 2[/u] [b]p6.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company will lay lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses. The Edison lighting company will hang strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [b]p7.[/b] You are given a sequence of $16$ digits. Is it always possible to select one or more digits in a row, so that multiplying them results in a square number? [img]https://cdn.artofproblemsolving.com/attachments/d/1/f4fcda2e1e6d4a1f3a56cd1a04029dffcd3529.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a chessboard that is infinite in all directions. Is it possible to place an infinite number of queens on it so that on each horizontally, on each vertical and on each diagonal of both directions (i.e. on a set of cells located at an angle of $45^o$ or $135^o$ to the horizontal) was exactly one queen?

You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$? [i]Author: Alex Zhu[/i]