The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$, \[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \] Find $a_{2004}$.

$(a_n)$ is an arithmetic sequence, $(b_n)$ is a geometric sequence. If $b_1=a_1^2,b_2=a_2^2,b_3=a_3^2(a_1<a_2)$, and $\lim_{n\to\infty}(b_1+b_2+\cdots+b_n)=\sqrt2+1$, find $a_n$.

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

Prove that every infinite arithmetic progression $a$, $a+d$, $a+2d$,... where $a$ and $d$ are positive integers, contains infinte geometric progression $b$, $bq$, $bq^2$,... where $b$ and $q$ are also positive integers

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

When the mean, median, and mode of the list $10,2,5,2,4,2,x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? $\text{(A)}\ 3\qquad\text{(B)}\ 6 \qquad\text{(C)}\ 9 \qquad\text{(D)}\ 17\qquad\text{(E)}\ 20$

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

The cubic equation $x^3+2x-1=0$ has exactly one real root $r$. Note that $0.4<r<0.5$. (a) Find, with proof, an increasing sequence of positive integers $a_1 < a_2 < a_3 < \cdots$


such that \[\frac{1}{2}=r^{a_1}+r^{a_2}+r^{a_3}+\cdots.\] (b) Prove that the sequence that you found in part (a) is the unique increasing sequence with the above property.

Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.

In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is: $\text{(A)} \ 3 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \frac{27}{19} \qquad \text{(D)} \ \frac{13}9 \qquad \text{(E)} \ \frac{23}{38}$

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence having $ x_1=3 $ and defined as $ x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor , $ for every natural number $ n. $ Find all values $ m $ for which the terms $ x_m,x_{m+1},x_{m+2} $ are in arithmetic progression, where $ \lfloor\rfloor $ denotes the integer part.

In $\triangle ABC$, $A,B,C$ are arithmetic sequence, and $c-a$ is equal to height on side $BC$, then $\sin\frac{C-A}{2}=$________.

$m$ is an integer satisfying $m \ge 2024$ , $p$ is the smallest prime factor of $m$ , for an arithmetic sequence $\{a_n\}$ of positive numbers with the common difference $m$ satisfying : for any integer $1 \le i \le \frac{p}{2} $ , there doesn’t exist an integer $x , y \le \max \{a_1 , m\}$ such that $a_i=xy$ Try to proof that there exists a positive real number $c$ such that for any $ 1\le i \le j \le n $ , $gcd(a_i , a_j ) = c \times gcd(i , j)$

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

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?

A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3,4,6$, then: $ \textbf{(A)}\ S_4=20 \qquad\textbf{(B)}\ S_4=25\qquad\textbf{(C)}\ S_5=49\qquad\textbf{(D)}\ S_6=49\qquad\textbf{(E)}\ S_2=\frac12 S_4 $

Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.

What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 4006$

Decide if there are ten natural and distinct numbers $a_1,a_2,\ldots ,a_{10}$ such that: $\bullet$ Each of them is a power of a natural number with a natural exponent and greater than $1$. $\bullet$ The numbers $a_1,a_2,\ldots ,a_{10}$ form an arithmetic progression.

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.)

Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teahcers at school taught her about how a young Karl Gauss quickly computed the sum \[1+2+3+\cdots+98+99+100\] in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?

The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

If $x\neq y$ and the sequences $x,a_1,a_2,y$ and $x,b_1,b_2,b_3,y$ each are in arithmetic progression, then $(a_2-a_1)/(b_2-b_1)$ equals $\textbf{(A) }\frac{2}{3}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad \textbf{(E) }\frac{3}{2}$

Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?

By definition, $ r! \equal{} r(r \minus{} 1) \cdots 1$ and $ \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}$, where $ r,j,k$ are positive integers and $ k < j$. If $ \binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $ n > 3$, then $ n$ equals $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 11\qquad \textbf{(E)}\ 12$