An infinite sequence of real numbers $a_1,a_2,a_3,\dots$ is called $\emph{spooky}$ if $a_1=1$ and for all integers $n>1$, \[\begin{array}{c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c} na_1&+&(n-1)a_2&+&(n-2)a_3&+&\dots&+&2a_{n-1}&+&a_n&<&0,\\ n^2a_1&+&(n-1)^2a_2&+&(n-2)^2a_3&+&\dots&+&2^2a_{n-1}&+&a_n&>&0. \end{array}\]Given any spooky sequence $a_1,a_2,a_3,\dots$, prove that \[2013^3a_1+2012^3a_2+2011^3a_3+\cdots+2^3a_{2012}+a_{2013}<12345.\]

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u]Set 3[/u] [b]P3.11[/b] Find all possible values of $c$ in the following system of equations: $$a^2 + ab + c^2 = 31$$ $$b^2 + ab - c^2 = 18$$ $$a^2 - b^2 = 7$$ [b]P3.12 / R5.25[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of$ ABCE$? [b]P3.13[/b] Thomas has two distinct chocolate bars. One of them is $1$ by $5$ and the other one is $1$ by $3$. If he can only eat a single $1$ by $1$ piece off of either the leftmost side or the rightmost side of either bar at a time, how many different ways can he eat the two bars? [b]P3.14[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. The entire triangle is revolved about side $BC$. What is the volume of the swept out region? [b]P3.15[/b] Find the number of ordered pairs of positive integers $(a, b)$ that satisfy the equation $a(a -1) + 2ab + b(b - 1) = 600$. [u]Set 4[/u] [b]P4.16[/b] Compute the sum of the digits of $(10^{2017} - 1)^2$ . [b]P4.17[/b] A right triangle with area $210$ is inscribed within a semicircle, with its hypotenuse coinciding with the diameter of the semicircle. $2$ semicircles are constructed (facing outwards) with the legs of the triangle as their diameters. What is the area inside the $2$ semicircles but outside the first semicircle? [b]P4.18[/b] Find the smallest positive integer $n$ such that exactly $\frac{1}{10}$ of its positive divisors are perfect squares. [b]P4.19[/b] One day, Sambuddha and Jamie decide to have a tower building competition using oranges of radius $1$ inch. Each player begins with $14$ oranges. Jamie builds his tower by making a $3$ by $3$ base, placing a $2$ by $2$ square on top, and placing the last orange at the very top. However, Sambuddha is very hungry and eats $4$ of his oranges. With his remaining $10$ oranges, he builds a similar tower, forming an equilateral triangle with $3$ oranges on each side, placing another equilateral triangle with $2$ oranges on each side on top, and placing the last orange at the very top. What is the positive difference between the heights of these two towers? [b]P4.20[/b] Let $r, s$, and $t$ be the roots of the polynomial $x^3 - 9x + 42$. Compute the value of $(rs)^3 + (st)^3 + (tr)^3$. [u]Set 5[/u] [b]P5.21[/b] For all integers $k > 1$, $\sum_{n=0}^{\infty}k^{-n} =\frac{k}{k -1}$. There exists a sequence of integers $j_0, j_1, ...$ such that $\sum_{n=0}^{\infty}j_n k^{-n} =\left(\frac{k}{k -1}\right)^3$ for all integers $k > 1$. Find $j_{10}$. [b]P5.22[/b] Nimi is a triangle with vertices located at $(-1, 6)$, $(6, 3)$, and $(7, 9)$. His center of mass is tied to his owner, who is asleep at $(0, 0)$, using a rod. Nimi is capable of spinning around his center of mass and revolving about his owner. What is the maximum area that Nimi can sweep through? [b]P5.23[/b] The polynomial $x^{19} - x - 2$ has $19$ distinct roots. Let these roots be $a_1, a_2, ..., a_{19}$. Find $a^{37}_1 + a^{37}_2+...+a^{37}_{19}$. [b]P5.24[/b] I start with a positive integer $n$. Every turn, if $n$ is even, I replace $n$ with $\frac{n}{2}$, otherwise I replace $n$ with $n-1$. Let $k$ be the most turns required for a number $n < 500$ to be reduced to $1$. How many values of $n < 500$ require k turns to be reduced to $1$? [b]P5.25[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $I$ and $O$ be the incircle and circumcircle of $ABC$, respectively. The altitude from $A$ intersects $I$ at points $P$ and $Q$, and $O$ at point $R$, such that $Q$ lies between $P$ and $R$. Find $PR$. PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and R16-30 /P6-10/ P26-30 [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\cdots,a_n$ be a sequence of real numbers lying between $1$ and $-1$, i.e. $-1<a_i<1$, for $1\leq i \leq n$ and such that (i) $a_1+a_2+\cdots+a_n=0$ (ii) $a_1^2+a_2^2+\cdots+a_n^2=40$ Determine the smallest possible value of $n$

Let $\alpha$ and $\beta$ be nonnegative integers. Suppose the number of strictly increasing sequences of integers $a_0,a_1,\dots,a_{2014}$ satisfying $0 \leq a_m \leq 3m$ is $2^\alpha (2\beta + 1)$. Find $\alpha$. [i]Proposed by Lewis Chen[/i]

Define a sequence ${{a_n}}_{n\ge1}$ such that $a_1=1$ , $a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $gcd(m,a_n)\neq{1}$. Show that all positive integers occur in the sequence.

On a table there are several cards, some face up and others face down. The allowed operation is to choose 4 cards and turn them over. The goal is to get all the cards in the same state (all face up or all face down). Determine if the objective can be achieved through a sequence of permitted operations if initially there are: a) 101 cards face up and 102 face down; b) 101 cards face up and 101 face down.

An arithmetic sequence is a sequence of $(a_1, a_2, \dots, a_n) $ such that the difference between any two consecutive terms is the same. That is, $a_ {i + 1} -a_i = d $ for all $i \in \{1,2, \dots, n-1 \} $, where $d$ is the difference of the progression. A sequence $(a_1, a_2, \dots, a_n) $ is [i]tlaxcalteca [/i] if for all $i \in \{1,2, \dots, n-1 \} $, there exists $m_i $ positive integer such that $a_i = \frac {1} {m_i}$. A taxcalteca arithmetic progression $(a_1, a_2, \dots, a_n )$ is said to be [i]maximal [/i] if $(a_1-d, a_1, a_2, \dots, a_n) $ and $(a_1, a_2, \dots, a_n, a_n + d) $ are not Tlaxcalan arithmetic progressions. Is there a maximal tlaxcalteca arithmetic progression of $11$ elements?

Let $a$ and $b$ be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence $a_n = a \cdot 2017^n + b \cdot 2016^n$ is infinite.

Show that there exists a positive integer $N$ such that for all integers $a>N$, there exists a contiguous substring of the decimal expansion of $a$, which is divisible by $2011$. Note. A contiguous substring of an integer $a$ is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of $a$.

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$? $\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$

Consider the recurrence relation $$a_{n+3}=\frac{a_{n+2}a_{n+1}-2}{a_n}$$ with initial condition $(a_0,a_1,a_2)=(1,2,5)$. Let $b_n=a_{2n}$ for nonnegative integral $n$. It turns out that $b_{n+2}+xb_{n+1}+yb_n=0$ for some pair of real numbers $(x,y)$. Compute $(x,y)$.

How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both? $ \textbf{(A)}\ 190\qquad\textbf{(B)}\ 192\qquad\textbf{(C)}\ 211\qquad\textbf{(D)}\ 380\qquad\textbf{(E)}\ 382$

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Define sequence $\{a_n\}$: $a_1$ is any positive integer, and for any positive integer $n\ge 1$, $a_{n+1}$ is the smallest positive integer coprime to $\sum_{i=1}^{n} a_i$ and not equal to $a_1,\ldots, a_n$. Prove that every positive integer appears in the sequence $\{a_n\}$.

Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold. [list] [*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$ [*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$. [/list]