The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and [*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume that $2n+1$ positive integers satisfy the following: If we remove any of these integers, the remaining $2n$ integers can be partitioned in two groups of $n$ numbers in each, such that the sum of the numbers in one group is equal to the sum of the numbers in the other. Prove that all of these numbers must be equal.

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.

Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose \[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\] Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$

Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$). Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$. [i](Salvatore Tringali)[/i]

Find all prime numbers $p$ for which there exist quadratic trinomials $P(x)$ and $Q(x)$ with integer coefficients, both with leading coefficients equal to $1$, such that the coefficients of $x^0$, $x^1$, $x^2$, and $x^3$ in the expanded form of the product $P(x)Q(x)$ are congruent modulo $p$ to $4$, $0$, $(-16)$, and $0$, respectively.

Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.

Determine whether or not there exist numbers $x_1,x_2,\ldots ,x_{2009}$ from the set $\{-1,1\}$, such that: \[x_1x_2+x_2x_3+x_3x_4+\ldots+x_{2008}x_{2009}+x_{2009}x_1=999\]

Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Let $n \geq 4$ be an integer. $a_1, a_2, \ldots, a_n \in (0, 2n)$ are $n$ distinct integers. Prove that there exists a subset of the set $\{a_1, a_2, \ldots, a_n \}$ such that the sum of its elements is divisible by $2n.$

[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? [b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand? [b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders. [b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Given a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Let $a_1$, $a_2$, $a_3$, $...$ denote the strictly increasing sequence of all positive integers $n$ such that $s(7n + 1) = 7s(n) + 1$. Compute $a_{2023}$.

Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

A sequence $\{y_i\}$ is given, where $y_0=-\frac{1}{4},y_1=0$. For every positive integer $n$ the following equality holds: $$y_{n-1}+y_{n+1}=4y_n+1$$ Prove that for every positive integer $n$ the number $2y_{2n}+\frac{3}{2}$ a) is a positive integer b) is a square of a positive integer [i]D. Zmiaikou[/i]

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

Prove that each positive integer is equal to a difference of two positive integers with the same number of the prime divisors.

From the number $7^{1996}$ we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits.

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72. a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$. b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $E(9,6,m_0)$.

Prove that a)$\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0$ (mod $2017^2$ ) b)$\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )$ (mod $2017^2$ )

Let $n$ be the least positive integer for which $149^n - 2^n$ is divisible by $3^3 \cdot 5^5 \cdot 7^7$. Find the number of positive divisors of $n$.