Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$. Let $x_{2021} = \frac{a}{b}$, where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$, then either $3|p-1$ or $p=3$. [i]Authored by Nikola Velov[/i]

Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows: [list=a] [*] $a_{10}$ is some positive integer, which can of course be written in base 10. [*] For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$. [/list] For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$. Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?

In a circular circuit there are petrol stations, so that the total accumulated petrol in them it is exactly enough for a car to go around the circuit. Prove that there is a position from where a car, with the tank of finite capacity and initially empty, can leave and get to go a full loop around the circuit, stopping to refuel at positions. [hide=original wording]En un circuito circular hay puestos de gasolina, de modo que el total de la gasolina acumulada en ellos es exactamente suciente para que un auto de una vuelta completa al circuito. Demostrar que existe un puesto desde donde un auto, con el estanque de capacidad finita e inicialmente vacio, puede partir y conseguir recorrer una vuelta completa al circuito, deteniendose a reabastecerse de gasolina en los puestos.[/hide]

A college student drove his compact car $ 120$ miles home for the weekend and averaged $ 30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $ 20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? $ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 28$

In the quadrilateral $ABCD$, the points $E, F$, and $K$ are midpoints of the $AB, BC, AD$ respectively. Known that $KE \perp AB, K F \perp BC$, and the angle $\angle ABC = 118^o$. Find $ \angle ACD$ (in degrees).

Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? $\textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{6}\qquad \textbf{(C)}\ \frac{11}{60}\qquad \textbf{(D)}\ \frac{1}{5}\qquad \textbf{(E)}\ \frac{7}{30}$

Consider three pairwise intersecting circles $\omega_1$, $\omega_2$ and $\omega_3$. Let their three common chords intersect at point $R$. We denote by $O_1$ the center of the circumcircle of a triangle formed by some triple common points of $\omega_1$ & $\omega_2$, $\omega_2$ & $\omega_3$ and $\omega_3$ & $\omega_1$. and we denote by $O_2$ the center of the circumcircle of the triangle formed by the second intersection points of the same pairs of circles. Prove that points $R$, $O_1$ and $O_2$ are collinear.

Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.

Spencer eats ice cream in a right circular cone with an opening of radius $5$ and a height of $10$. If Spencer’s ice cream scoops are always perfectly spherical, compute the radius of the largest scoop he can get such that at least half of the scoop is contained within the cone.

We call $\triangle{ABC}$ with centroid $G$ [i]balanced[/i] on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B.$ $\triangle{XYZ},$ with $XY=2023$ and $\angle{ZXY}=120^\circ,$ is balanced on $XY.$ What is the maximum value of $XZ$?

Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?

The Professor chooses to assign homework problems from a set of problems labeled $1$ to $100$, inclusive. He will not assign two problems whose numbers share a common factor greater than $1$. If the Professor chooses to assign the maximum number of homework problems possible, how many different combinations of problems can he assign?

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ rows and $ n$ columns , the percentage of $ 1$s on that row or column is not less than $ a$ or greater than $ 100\minus{}a$. Find the largest beautiful number.

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Let $P$ be a point on side $\overline{AB}$ of equilateral triangle $ABC$. If $BP = 6$ and $CP = 9$, what is the length of $AB$? $\textbf{(A) }2\sqrt5\qquad\textbf{(B) }3+\sqrt6\qquad\textbf{(C) }3\sqrt5\qquad\textbf{(D) }3\sqrt6 + 3\qquad\textbf{(E) }6\sqrt2$

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with \$256, find the probability that after 8 bets, he will have at least \$50. [i]Proposed by Jeffrey Tong[/i]

Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\] (A triangle is non-degenerate if its vertices are not collinear.) [i]Proposed by Bruno Le Floch, France[/i]

Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB,$ and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC.$ Suppose that $BC = 27, CD = 25,$ and $AP = 10.$ If $MP = \tfrac {a}{b}$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.

The fields of an $n\times n$ chess board are colored black and white, such that in every small $2\times 2$-square both colors should be the same number. How many there possibilities are for this?

Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that \begin{align*} cy-bz &\ge 0, \\ az-cx &\ge 0, \\ bx-ay &\ge 0. \end{align*}