A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID? [i]Proposed by Richard Chen[/i]

A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\] where $n$ and $k$ are positive integers.

Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.

Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then, [b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$ [b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$

Is there a function $f(x)$ defined and continuous on $R$ such that: a) $f(f(x)) = 1 + 2x$ ? b) $f(f(x)) = 1 - 2x $?

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$  at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects  $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$. [i]Proposed by Ali Zamani [/i]

There exist complex numbers $z_1,z_2,\dots,z_{10}$ which satisfy$$|z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}|$$for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. If $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1|+|z_2|+\cdots+|z_{10}|$. [i]Proposed by DeToasty3[/i]

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy] $ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

There are given integers $a$ and $b$ such that $a$ is different from $0$ and the number $3+ a +b^2$ is divisible by $6a$. Prove that $a$ is negative.

Let $F$, $D$, and $E$ be points on the sides $[AB]$, $[BC]$, and $[CA]$ of $\triangle ABC$, respectively, such that $\triangle DEF$ is an isosceles right triangle with hypotenuse $[EF]$. The altitude of $\triangle ABC$ passing through $A$ is $10$ cm. If $|BC|=30$ cm, and $EF \parallel BC$, calculate the perimeter of $\triangle DEF$.

[b]p9.[/b] The frozen yogurt machine outputs yogurt at a rate of $5$ froyo$^3$/second. If the bowl is described by $z = x^2+y^2$ and has height $5$ froyos, how long does it take to fill the bowl with frozen yogurt? [b]p10.[/b] Prankster Pete and Good Neighbor George visit a street of $2021$ houses (each with individual mailboxes) on alternate nights, such that Prankster Pete visits on night $1$ and Good Neighbor George visits on night $2$, and so on. On each night $n$ that Prankster Pete visits, he drops a packet of glitter in the mailbox of every $n^{th}$ house. On each night $m$ that Good Neighbor George visits, he checks the mailbox of every $m^{th}$ house, and if there is a packet of glitter there, he takes it home and uses it to complete his art project. After the $2021^{th}$ night, Prankster Pete becomes enraged that none of the houses have yet checked their mail. He then picks three mailboxes at random and takes out a single packet of glitter to dump on George’s head, but notices that all of the mailboxes he visited had an odd number of glitter packets before he took one. In how many ways could he have picked these three glitter packets? Assume that each of these three was from a different house, and that he can only visit houses in increasing numerical order. [b]p11. [/b]The taxi-cab length of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $|x_1 - x_2| + |y_1- y_2|$. Given a series of straight line segments connected head-to-tail, the taxi-cab length of this path is the sum of the taxi-cab lengths of its line segments. A goat is on a rope of taxi-cab length $\frac72$ tied to the origin, and it can’t enter the house, which is the three unit squares enclosed by $(-2, 0)$,$(0, 0)$,$(0, -2)$,$(-1, -2)$,$(-1, -1)$,$(-2, -1)$. What is the area of the region the goat can reach? (Note: the rope can’t ”curve smoothly”-it must bend into several straight line segments.) [b]p12.[/b] Parabola $P$, $y = ax^2 + c$ has $a > 0$ and $c < 0$. Circle $C$, which is centered at the origin and lies tangent to $P$ at $P$’s vertex, intersects $P$ at only the vertex. What is the maximum value of a, possibly in terms of $c$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A pair (a,b) of positive integers is [i]Rioplatense [/i]if it is true that $b + k$ is a multiple of $a + k$ for all $k \in\{ 0 , 1 , 2 , 3 , 4 \}$. Prove that there is an infinite set $A$ of positive integers such that for any two elements $a$ and $b$ of $A$, with $a < b$, the pair $(a,b)$ is [i]Rioplatense[/i].

Let $n = 2^{19}3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$.What is the number formed by taking the last two digits of $M$ (in the same order)?

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]