An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than $100$. (b) Give an example of such a progression with $72$ terms. (c) Prove that the number of terms in any such progression does not exceed $72$. (V. Bugaenko and Tarasov, Moscow)

Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

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 $p$.

Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

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 $x_1<y_1<x_2<y_2<...<x_n<y_n$

Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)

Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

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.

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

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]

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.