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]

Source: 1976 Euclid Part B Problem 2 ----- Given that $x$, $y$, and $2$ are in geometric progression, and that $x^{-1}$, $y^{-1}$, and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$.

Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\]

Let $a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci}, \overline{dea}, \overline{ifd}, \overline{jgf}, \overline{bfeg},\ldots$ be an increasing arithmetic progression. Find the $16$th term of this sequence.

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 $M$.

We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$. Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.

The first item and common difference of an arithmetic sequence are nonnegative intengers. The number of items is not less than $3$, and the sum of all items is $97^2$. Then the number of such sequences is $\text{(A)}2\qquad\text{(B)}3\qquad\text{(C)}4\qquad\text{(D)}5$

An [i]arithmetic sequence[/i] is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$. A [i]geometric sequence[/i] is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$. [list=a] [*] Does every arithmetic sequence of [b]integers[/b] have an infinite subsequence which is geometric? [*] Does every arithmetic sequence of [b]real numbers[/b] have an infinite subsequence which is geometric? [/list]

Find the eighth term of the sequence $1440,$ $1716,$ $1848,\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.

Let $n$ be a positive integer greater than or equal to $6$, and suppose that $a_1, a_2, ...,a_n$ are real numbers such that the sums $a_i + a_j$ for $1 \le i<j\le n$, taken in some order, form consecutive terms of an arithmetic progression $A$, $A + d$, $...$ ,$A + (k-1)d$, where $k = n(n-1)/2$. What are the possible values of $d$?

Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that \[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\] where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.

For $n=2^m$ (m is a positive integer) consider the set $M(n) = \{ 1,2,...,n\}$ of natural numbers. Prove that there exists an order $a_1, a_2, ..., a_n$ of the elements of M(n), so that for all $1\leq i < j < k \leq n$ holds: $a_j - a_i \neq a_k - a_j$.

Suppose that $ \{a_n\}$ is an arithmetic sequence with \[a_1 \plus{} a_2 \plus{} \cdots \plus{} a_{100} \equal{} 100\quad\text{and}\quad a_{101} \plus{} a_{102} \plus{} \cdots \plus{} a_{200} \equal{} 200.\] What is the value of $ a_2 \minus{} a_1$? $ \textbf{(A)}\ 0.0001 \qquad \textbf{(B)}\ 0.001 \qquad \textbf{(C)}\ 0.01 \qquad \textbf{(D)}\ 0.1 \qquad \textbf{(E)}\ 1$

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

Let $\{a_n\}_{n\geq 0}$ be an arithmetic sequence with difference $d$ and $1\leq a_0\leq d$. Denote the sequence as $S_0$, and define $S_n$ recursively by two operations below: Step $1$: Denote the first number of $S_n$ as $b_n$, and remove $b_n$. Step $2$: Add $1$ to the first $b_n$ numbers to get $S_{n+1}$. Prove that there exists a constant $c$ such that $b_n=[ca_n]$ for all $n\geq 0$, where $[]$ is the floor function.

$(a_n)$ 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 $a_n$.

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Let be a bipartition of the set formed by the first $ 13 $ nonnegative numbers. Prove that at least one of these two subsets that form this partition contains an arithmetic progression.

Let positive numbers $a_1, a_2, ..., a_{3n}$ $(n \geq 2)$ constitute an arithmetic progression with common difference $d > 0$. Prove that among any $n + 2$ terms in this progression, there exist two terms $a_i, a_j$ $(i \neq j)$ satisfying $1 < \frac{|a_i - a_j|}{nd} < 2$.

Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$. [i]Proposed by Jack Cornish[/i]

For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.

The function $f:\mathbb{N}\rightarrow \mathbb{N}$ is [b]peruvian[/b] if it satifies the following two properties: $\triangleright f$ is strictly increasing. $\triangleright$ The numbers $a_1,a_2,a_3,\dots$ where $a_1=f(1)$ and $a_{n+1}=f(a_n)$ for every $n\geq 1$, are in arithmetic progression. Determine all peruvian functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(1)=3$.

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?