At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.

Let $ABC$ be a triangle and let $M$ be the midpoint $BC$. On the exterior of the triangle, consider the parallelogram $BCDE$ such that $BE//AM$ and $BE=AM/2$ . Prove that line $EM$ passes through the midpoint of segment $AD$.

Let $n>1$ be a positive integer. Each unit square in an $n\times n$ grid of squares is colored either black or white, such that the following conditions hold: $\bullet$ Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; $\bullet$ Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; $\bullet$ Any $2\times 2$ subgrid contains at least one square of each color. Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).

Define $x\star y=\frac{\sqrt{x^2+3xy+y^2-2x-2y+4}}{xy+4}$. Compute \[((\cdots ((2007\star 2006)\star 2005)\star\cdots )\star 1).\]

The trapeze $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle $\Gamma$. Let $X$ be a variable point of the arc $\overarc{AB}$ that does not contain either $C$ or $D$. Let $Y$ be the point of intersection of $AB$ and $DX$, and let $Z$ be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$. Prove that the measure of the angle $\angle AZX$ does not depend on the choice of $X$.

Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide. [i]Author: Anderson Wang[/i]

A triangle $ABC$ is given. For every positive numbers $p,q,r$, let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC} $, and $\overrightarrow{AC'}=r\overrightarrow{CA}$. Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$. Prove that for all positive numbers $x,y,z$ and every positive integer $n$, $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$.

A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$.

A positive integer $b\geq 2$ is [i]neat[/i] if and only if there exist positive base-$b$ digits $x$ and $y$ (that is, $x$ and $y$ are integers, $0<x<b$ and $0<y<b$) such that the number $x.y$ base $b$ (that is, $x+\tfrac yb$) is an integer multiple of $x/y$. Find the number of [i]neat[/i] integers less than or equal to $100$.

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

A deck consists of $n$ different cards. A move consists of taking out a group of cards in sequence from some place in the deck, and putting it back someplace else without changing the order within the group or turning any cards over. We are required to reverse the order of cards in the deck by such moves. (a) Prove that for $n = 9$, this can be done in $5$ moves. (b) Prove that for $n = 52$, this i. can be done in $27$ moves, ii. can’t be done in $17$ moves, iii. can’t be done in $26$ moves. (SM Voronin, Tchelyabinsk)

Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$; b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$. Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if \[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\] [i]Evan O'Dorney.[/i]

The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find \[\lim_{n\to\infty}f(n)^{1/n}\]

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second. [img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]

In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$. [i]Proposed by Vivian Loh [/i]

Austin and Temple are $ 50$ miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging $ 60$ miles per hour. Leaving the car with her daughter, Bonnie rod a bus back to Austin along the same route and averaged $ 40$ miles per hour on the return trip. What was the average speed for the round trip, in miles per hour? $ \textbf{(A)}\ 46 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 54$

The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$. [i]Proposed by Michael Tang[/i]

Point $P$ is inside triangle $\triangle{ABC}$ such that $\angle{ABP}=\angle{ACP}.$ Given that $AB=6, AC=8, BC=7,$ and $\tfrac{BP}{PC}=\tfrac{1}{2},$ compute $\tfrac{[BPC]}{[ABC]}.$ (Here, $[XYZ]$ denotes the area of $\triangle{XYZ}.$)

A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds: For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials. $a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$. $b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials. [i]Proposed by Alireza Shavali[/i]

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Let $ABC$ be a triangle with $AB>AC$. Let $D$ be a point on side $AB$ such that $BD=AC$. Consider the circle $\gamma$ passing through point $D$ and tangent to side $AC$ at point $A$. Consider the circumscribed circle $\omega$ of the triangle $ABC$ that interesects the circle $\gamma$ at points $A$ and $E$. Prove that point $E$ is the intersection point of the perpendicular bisectors of line segments $BC$ and $AD$.

Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.

[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.