Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

An arithmetic sequence of five terms is considered $good$ if it contains 19 and 20. For example, $18.5,19.0,19.5,20.0,20.5$ is a $good$ sequence. For every $good$ sequence, the sum of its terms is totalled. What is the total sum of all $good$ sequences?

Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?

Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.

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 $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

Let $a_{1},a_{2},\ldots$ be an infinite arithmetic progression. It's known that there exist positive integers $p,q,t$ such that $a_{p}+tp=a_{q}+tq$. If $a_{t}=t$ and the sum of the first $t$ numbers in the sequence is $18$, determine $a_{2008}$.

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$. Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

Let $S$ be a set of $6$ positive real numbers such that $\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S$ or $a-b \in S$ Prove that if we sort these numbers in ascending order, then they form an arithmetic progression

Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.

The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.

Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$

For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.

The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.

You are given $n\ge 4$ positive real numbers. It turned out that all $\frac{n(n-1)}{2}$ of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal. [i](Proposed by Anton Trygub)[/i]

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

An arithmetic progression is formed by $9$ positive integers such that the product of these $9$ terms is a multiple of $3$. Prove that said product is also multiple of $81$.

Let $P_1,P_2,\ldots ,P_n$, be infinite arithmetic progressions of positive integers, of differences $d_1,d_2,\ldots ,d_n$, respectively. Prove that if every positive integer appears in at least one of the $n$ progressions then one of the differences $d_i$ divides the least common multiple of the remaining $n-1$ differences. Note: $P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}$ with $ a_i$ and $d_i$ positive integers.

Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .

Let $a_1, a_2, a_3,\ldots$ and $b_1, b_2, b_3,\ldots$ be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio $a_k/b_k$ is an integer for all k. Is it true that this ratio does not depend on $k{}$? [i]Boris Frenkin[/i]

Determine all integers $n\geqslant 3$ such that there exist $n{}$ pairwise distinct real numbers $a_1,\ldots,a_n$ for which the sums $a_i+a_j$ over all $1\leqslant i<j\leqslant n$ form an arithmetic progression.

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?