Let $(a_n)_{n\geq 0}$ be a sequence defined by $a_0\geq 0$ and the recurrence relation $$a_{n+1}=\frac{a_n^2-1}{n+1},$$ for all $n\geq 0$. Prove that here exists a real number $a> 0$ such that: [list] [*] if $a_0\geq a,$ $\lim_{n\rightarrow\infty}a_n = \infty$; [*] if $a_0\in [0,a),$ $\lim_{n\rightarrow\infty}a_n = 0$.

We are given the sequence $a_1,a_2,a_3,\ldots$, for which: $$a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.$$ Prove that the numbers $a_1$, $a_2$ and $\frac{a_1^2+a_2^2+c}{a_1a_2}$ are whole numbers.

You own a calculator that computes exactly. It has all the standard buttons, including a button that replaces the number currently displayed with its arctangent, and a button that replaces whatever is currently displayed with its cosine. You turn on the calculator and it reads 0. You create a sequence by alternately clicking on the arctangent button and the cosine button. (The calculator is in radian mode.) Let $a_n$ be the value displayed after you've pressed the cosine button for the $n$th time. What is $\prod_{k=1}^{11} a_k$?

The first three terms of an arithmetic sequence are $ 2x\minus{}3$, $ 5x\minus{}11$, and $ 3x\plus{}1$ respectively. The $ n$th term of the sequence is $ 2009$. What is $ n$? $ \textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$

Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$. Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.

A sequence of positive integers $a_1, a_2,\ldots, a_n$ is called [i]ladder[/i] of length $n$ if it consists of $n$ consecutive integers in ascending order. (a) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common, $a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is equal to $1$. (b) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common, $a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is greater than $1$.

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

[u]Round 1[/u] [b]p1.[/b] If $2m = 200 cm$ and $m \ne 0$, find $c$. [b]p2.[/b] A right triangle has two sides of lengths $3$ and $4$. Find the smallest possible length of the third side. [b]p3.[/b] Given that $20(x + 17) = 17(x + 20)$, determine the value of $x$. [u]Round 2[/u] [b]p4.[/b] According to the Egyptian Metropolitan Culinary Community, food service is delayed on $\frac23$ of flights departing from Cairo airport. On average, if flights with delayed food service have twice as many passengers per flight as those without, what is the probability that a passenger departing from Cairo airport experiences delayed food service? [b]p5.[/b] In a positive geometric sequence $\{a_n\}$, $a_1 = 9$, $a_9 = 25$. Find the integer $k$ such that $a_k = 15$ [b]p6.[/b] In the Delicate, Elegant, and Exotic Music Organization, pianist Hans is selling two types of owers with different prices (per ower): magnolias and myosotis. His friend Alice originally plans to buy a bunch containing $50\%$ more magnolias than myosotis for $\$50$, but then she realizes that if she buys $50\%$ less magnolias and $50\%$ more myosotis than her original plan, she would still need to pay the same amount of money. If instead she buys $50\%$ more magnolias and $50\%$ less myosotis than her original plan, then how much, in dollars, would she need to pay? [u]Round 3[/u] [b]p7.[/b] In square $ABCD$, point $P$ lies on side $AB$ such that $AP = 3$,$BP = 7$. Points $Q,R, S$ lie on sides $BC,CD,DA$ respectively such that $PQ = PR = PS = AB$. Find the area of quadrilateral $PQRS$. [b]p8.[/b] Kristy is thinking of a number $n < 10^4$ and she says that $143$ is one of its divisors. What is the smallest number greater than $143$ that could divide $n$? [b]p9.[/b] A positive integer $n$ is called [i]special [/i] if the product of the $n$ smallest prime numbers is divisible by the sum of the $n$ smallest prime numbers. Find the sum of the three smallest special numbers. [u]Round 4[/u] [b]p10.[/b] In the diagram below, all adjacent points connected with a segment are unit distance apart. Find the number of squares whose vertices are among the points in the diagram and whose sides coincide with the drawn segments. [img]https://cdn.artofproblemsolving.com/attachments/b/a/923e4d2d44e436ccec90661648967908306fea.png[/img] [b]p11.[/b] Geyang tells Junze that he is thinking of a positive integer. Geyang gives Junze the following clues: $\bullet$ My number has three distinct odd digits. $\bullet$ It is divisible by each of its three digits, as well as their sum. What is the sum of all possible values of Geyang's number? [b]p12.[/b] Regular octagon $ABCDEFGH$ has center $O$ and side length $2$. A circle passes through $A,B$, and $O$. What is the area of the part of the circle that lies outside of the octagon? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2936505p26278645]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.

An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+\dfrac{1}{a_3a_4}\cdots=\dfrac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$. [i]Individual #7[/i]

Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold: 1. $a_n\neq 0$; 2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$. [i] Proposed by usjl[/i]