Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

The set of all finite ordered sets of $0$ and $ 1$ is somehow partitioned into two disjoint classes. Prove that any infinite sequence of $0$ and $1$ can be cut into non-intersecting finite parts such that all of these parts (except perhaps the first) belong to the same class.

An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?

Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$ Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

The sequence $(a_n)$ of real numbers is defined as follows: \[a_1=1, \qquad a_2=2, \quad \text{and} \quad a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3.\] Prove that for $n \geq 3$, $a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1$, where $[x]$ denotes the integer $p$ such that $p \leq x < p + 1$.

[u]Round 1[/u] [b]p1.[/b] Elaine creates a sequence of positive integers $\{s_n\}$. She starts with $s_1 = 2018$. For $n \ge 2$, she sets $s_n =\frac12 s_{n-1}$ if $s_{n-1}$ is even and $s_n = s_{n-1} + 1$ if $s_{n-1}$ is odd. Find the smallest positive integer $n$ such that $s_n = 1$, or submit “$0$” as your answer if no such $n$ exists. [b]p2.[/b] Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins? [b]p3.[/b] Four circles each of radius $\frac14$ are centered at the points $\left( \pm \frac14, \pm \frac14 \right)$, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle? [u]Round 2 [/u] [b]p4.[/b] If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing? [b]p5.[/b] Let $a_1 = 2018$, and for $n \ge 2$ define $a_n = 2018^{a_{n-1}}$ . What is the ones digit of $a_{2018}$? [b]p6.[/b] We can write $(x + 35)^n =\sum_{i=0}^n c_ix^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$? [u]Round 3[/u] [b]p7.[/b] How many positive integers are factors of $12!$ but not of $(7!)^2$? [b]p8.[/b] How many ordered pairs $(f(x), g(x))$ of polynomials of degree at least $1$ with integer coefficients satisfy $f(x)g(x) = 50x^6 - 3200$? [b]p9.[/b] On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between $1$ and $10$ (inclusive). What is the probability that the average of their three scores is an integer? [u]Round 4[/u] [b]p10.[/b] Find the largest positive integer N such that $$(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)$$ is divisible by $N$ for all choices of positive integers $a > b > c > d > e$. [b]p11.[/b] Let $ABCDE$ be a square pyramid with $ABCD$ a square and E the apex of the pyramid. Each side length of $ABCDE$ is $6$. Let $ABCDD'C'B'A'$ be a cube, where $AA'$, $BB'$, $CC'$, $DD'$ are edges of the cube. Andy the ant is on the surface of $EABCDD'C'B'A'$ at the center of triangle $ABE$ (call this point $G$) and wants to crawl on the surface of the cube to $D'$. What is the length the shortest path from $G$ to $D'$? Write your answer in the form $\sqrt{a + b\sqrt3}$, where $a$ and $b$ are positive integers. [b]p12.[/b] A six-digit palindrome is a positive integer between $100, 000$ and $999, 999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a,n_k = b$, and $(n_i + n_{i+1}) | n_in_{i+1}$ for all $i = 1,2,..., k - 1$

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$ $\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$

Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.

Derek the Dolphin and Kevin the Frog are playing a game where they take turns taking coins from a stack of $N$ coins, except with one rule: The number of coins someone takes each turn must be a power of $6$. The person who cannot take any more coins loses. If Derek goes first, how many integers $N$ from $1$ to $6^{2021}$ inclusive will guarantee him a win? (Example: If $N = 37$, then a possible sequence of turns is: Derek takes one coin, Kevin takes $36$ coins, and Kevin wins.)

Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$. [i]Proposed by Tristan Shin[/i]

Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)

A strictly increasing sequence $\{x_i\}_{i=1}^{\infty}$ of positive integers is said to be [i]large[/i] if, for every real number $L$, there exists an integer $n$ such that $\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} > L$. Do there exist large sequences $\{a_i\}_{i=1}^\infty$ and $\{b_i\}_{i=1}^{\infty}$ such that the sequence $\{a_i+b_i\}_{i=1}^{\infty}$ is not large? [i]Proposed by Lewis Chen[/i]