Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.

Given an acute $\vartriangle ABC$ with incenter $I$. Let $I'$ be the symmetric point $I$ with respect to the midpoint of $B,C$ and $D$ is the foot of $A$. If $DI$ and the circumcircle of vartriangle $BI'C$ intersect at $T$ and $TI' $ intersects the circumcircle of $\vartriangle ATI$ at $X$. Furthermore, $E,F$ are tangent points of the incircle and $AB,AC, P$ is the another intersection of the circumcircles of $\vartriangle ABC, \vartriangle AEF$. Show that $AX \parallel PI$. [img]https://3.bp.blogspot.com/-tj9A8HIR6Vw/XndLEPMRvnI/AAAAAAAALfk/2vw_pZbhpnkTKIc1BcKf4K7SNZ11vu4TACK4BGAYYCw/s1600/2018%2Bimoc%2Bg4.png[/img]

There is a right triangle $\triangle ABC$ in which $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ that satisfy $\angle ACX=\angle XCY=\angle YCZ=\angle ZCB$ and $BZ=2AX$. The smallest angle of $\triangle ABC$ is $\tfrac ab$ degrees, where $a,b$ are positive integers such that $\gcd(a,b)=1$. Find $a+b$.

Let $P(x)=x^3+x^2-r^2x-2020$ be a polynomial with roots $r,s,t$. What is $P(1)$? [i]Proposed by James Lin.[/i]

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

In $ xy \minus{}$plane, there are $ b$ blue and $ r$ red rectangles whose sides are parallel to the axis. Any parallel line to the axis can intersect at most one rectangle with same color. For any two rectangle with different colors, there is a line which is parallel to the axis and which intersects only these two rectangles. $ (b,r)$ cannot be ? $\textbf{(A)}\ (1,7) \qquad\textbf{(B)}\ (2,6) \qquad\textbf{(C)}\ (3,4) \qquad\textbf{(D)}\ (3,3) \qquad\textbf{(E)}\ \text{None}$

What is the area of a square in square feet, if each of its diagonals is $4$ feet long?

Right triangle $ABC$ has its right angle at $A$. A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$. Let $D$ be the point where the semicircle is tangent to $BC$. If $AD=4$ and $CO=5$, find $\cos\angle{ABC}$. [asy] import olympiad; pair A, B, C, D, O; A = (1,0); B = origin; C = (1,1); O = incenter(C, B, (1,-1)); draw(A--B--C--cycle); dot(O); draw(arc(O, 0.41421356237,0,180)); D = O+0.41421356237*dir(135); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,NW); label("$O$",O,S); [/asy] $\text{(A) }\frac{\sqrt{5}}{4}\qquad\text{(B) }\frac{3}{5}\qquad\text{(C) }\frac{12}{25}\qquad\text{(D) }\frac{4}{5}\qquad\text{(E) }\frac{2\sqrt{5}}{5}$

The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$. The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$. Prove that there is a natural number, that doesn't belong to $T_{1987}$.

Let $a, b, c$ be integers. Prove that there exist integers $p_1, q_1, r_1, p_2, q_2$ and $r_2$, satisfying $a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1$ and $c = p_1q_2 - p_2q_1.$

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

There are $n$ students. Denoted the number of the selections to select two students (with their weights are $a$ and $b$) such that $\left| {a - b} \right| \le 1$ (kg) and $\left| {a - b} \right| \le 2$ (kg) by $A_1$ and $A_2$, respectively. Prove that $A_2<3A_1+n$.

A rectangular field is half as wide as it is long and is completely enclosed by $ x$ yards of fencing. The area in terms of $ x$ is: $ \textbf{(A)}\ \frac {x^2}{2} \qquad\textbf{(B)}\ 2x^2 \qquad\textbf{(C)}\ \frac {2x^2}{9} \qquad\textbf{(D)}\ \frac {x^2}{18} \qquad\textbf{(E)}\ \frac {x^2}{72}$

Quadrilateral $ABCD$ is inscribed in a circle with $\angle ABC=60^\circ$ and $BC=CD$. Prove that $AB=AD+DC$.

Let $n$ be a natural number and $f(x)$ be a polynomial with real coefficients having $n$ different positive real roots. Is it possible the polynomial: $$x(x+1)(x+2)(x+4)f(x)+a$$ to be presented as the $k$-th power of a polynomial with real coefficients, for some natural $k\geq 2$ and real $a$?

[u]Round 1[/u] [b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment? [b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img] [b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$? [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2[/u] [b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them. [b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if [b](a)[/b] $n=2000;$ [b](b)[/b] $n=2001 \ ?$

Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$. [i]B. Berceanu[/i]

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that for all $x$ with $0<|x|\le c$ and for all $\alpha$ with $0<\alpha<1$, we have: $$a^\alpha b^{1-\alpha}\le\frac{a\sinh\alpha x}{\sinh x}+\frac{b\sinh x(1-\alpha)}{\sinh x}.$$

10. Let $n \ge 5$ be an odd number and let $r$ be an integer such that $1\le r \le (n-1)/2$. IN a sports tournament, $n$ players take part in a series of contests. In each contest, $2r+1$ players participate, and the scores obtained by the players are the numbers $$-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r$$ in some order. Each possible subset of $2r+1$ players takes part together in exactly one contest. let the final score of player $i$ be $S_i$, for each $i=1, 2,\cdots,n$. Define $N$ to be the smallest difference between the final scores of two players, i.e., $$N = \min_{i<j}|S_i - S_j|.$$ Determine, with proof, the maximum possible value of $N$.

Find all triples $(k, m, n)$ of positive integers such that $m$ is a prime and: (1) $kn$ is a perfect square; (2) $\frac{k(k-1)}{2}+n$ is a fourth power of a prime; (3) $k-m^2=p$ where $p$ is a prime; (4) $\frac{n+2}{m^2}=p^4$.

The real factors of $ x^2\plus{}4$ are: $\textbf{(A)}\ (x^2+2)(x^2+2) \qquad \textbf{(B)}\ (x^2+2)(x^2-2) \qquad \textbf{(C)}\ x^2(x^2+4) \qquad\\ \textbf{(D)}\ (x^2-2x+2)(x^2+2x+2) \qquad \textbf{(E)}\ \text{Non-existent}$

Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that $P$ externally touches on $A, B, Q$ and $S$, $Q$ externally touches on $P, B$ and $R$, $R$ externally touches on $A, B, Q$ and $S$, $S$ externally touches on $P, A$ and $R$. Calculate the length of $r.$ [asy] unitsize(0.3 cm); pair A, B, P, Q, R, S; real r = (3 + sqrt(17))/2; A = (-1,0); B = (1,0); P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180)); R = -P; Q = (r + 2,0); S = (-r - 2,0); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(P,r)); draw(Circle(Q,r)); draw(Circle(R,r)); draw(Circle(S,r)); label("$A$", A); label("$B$", B); label("$P$", P); label("$Q$", Q); label("$R$", R); label("$S$", S); [/asy]

Let $m,k$ be positive integers, $k<m$ and $M$ a set with $m$ elements. Prove that the maximal number of subsets $A_1,A_2,\ldots ,A_p$ of $M$ for which $A_i\cap A_j$ has at most $k$ elements, for every $1\le i<j\le p$, equals \[ p_{max}=\binom{m}{0}+\binom{m}{1}+\binom{m}{2}+\ldots+\binom{m}{k+1}\]

Non-negative integers are placed on the vertices of a $100$-gon, the sum of the numbers is $99$. Every minute at one of the vertices that is equal to $0$ will be replaced by $2$ and both its neighboring numbers are subtracted by $1$. Prove that after a while a negative number will appear on the board.