Let $a_n$ be a sequence defined as $a_1=1$, $a_2=2$, and $a_n=a_{n-1}-a_{n-2}$. Compute $a_{2015}$.

Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ via $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. What is the smallest $N$ such that $a_{N} > 2017$?

All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted: 1. Each person moves to the next seat clockwise. 2. Only the ones in seats 1 and $k$ exchange their seats. Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.

The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$, for all positive integers $n$. Find all $n$ for which $X^2+1\mid P_{n}(X)$

The following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 0; i < 12; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy] Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move. Let $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.

Let $m$ and $n$ be two natural numbers and let $d = gcd (m, n)$ their greatest common divisor. Let $a_1, a_2,...$ and $b_1, b_2, ...$ be two sequences of integers which are periodic with periods $m$ and $n$ respectively (this means that $a_{i + m} = a_i$ and $b_{i + n} = b_i$ for all natural numbers $i \ge 1$, note that there could be smaller periods). Prove that if the two sequences on the first $m + n - d$ terms match (i.e. $a_i = b_i$ for all $i \in \{1, 2, ..., m + n - d\}$), then they are the same (so $a_i = b_i$ for all natural $i \ge 1$).

Denote by $\lceil x \rceil$ the greatest integer less than or equal to $x$ and by $S(x)$ the sequence $\lceil x \rceil, \lceil 2x \rceil, \lceil 3x \rceil, \dots.$ Prove that there are distinct real solutions $\alpha$ and $\beta$ of the equation $$x^3-10x^2+29x-25=0$$ such that infinitely many positive integers appear both in $S(\alpha)$ and in $S(\beta).$

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$. (M Kontsevich, Moscow)

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the two sides of the road two lines of trees are planted. On every tree, the number of oaks among itself and its neighbors is written. (For the first and last trees, this is the number of oaks among itself and its only neighbor). Prove that if the two sequences of numbers on the trees are equal, then sequnces of trees on the two sides of the road are equal [I]Proposed by A. Khrabrov, D. Rostovski[/i]

Construct a holomorphic function $f(z) = \sum \limits_{n = 0} ^ \infty a_n z^n$ ( | z | <1 ) in the unit circle that can be analytically continued to all points of the unit circle except one point, and for which the sequence $\{a_n\}$ has two limit points, $\infty$ and a finite value.

Let $T=\text{TNFTPP}$. Three distinct positive Fibonacci numbers, all greater than $T$, are in arithmetic progression. Let $N$ be the smallest possible value of their sum. Find the remainder when $N$ is divided by $2007$.

Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.