Found problems: 93
Fractal Edition 1, P1
Show that any arithmetic progression where the first term and the common difference are non-zero natural numbers contains an infinite number of composite terms.
*A number is composite if it is not prime.
1964 Poland - Second Round, 3
Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.
2001 German National Olympiad, 1
Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression
2017 Singapore MO Open, 4
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 $x_1<y_1<x_2<y_2<...<x_n<y_n$
1980 IMO Shortlist, 13
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.
2024 Romania National Olympiad, 4
Let $a$ be a given positive integer. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n=\frac{1}{1+na},$ for every positive integer $n.$
Prove that for any integer $k \ge 3,$ there exist positive integers $n_1<n_2<\ldots<n_k$ such that the numbers $x_{n_1},x_{n_2},\ldots,x_{n_k}$ are consecutive terms in an arithmetic progression.
2022 Kyiv City MO Round 1, Problem 4
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]
1980 IMO Longlists, 13
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.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.8
Let's write down a segment of a series of integers from $0$ to $1995$. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining $1994$ numbers. Let $K$ be the length of the progression. Which two numbers must be crossed out so that the value of $K$ is the smallest?
2006 Grigore Moisil Urziceni, 2
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.
2011 IFYM, Sozopol, 4
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.
2018 Auckland Mathematical Olympiad, 4
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?
2020 Estonia Team Selection Test, 1
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
1986 Tournament Of Towns, (127) 2
Does there exist a number $N$ so that there are $N - 1$ infinite arithmetic progressions with differences $2 , 3 , 4 ,..., N$ , and every natural number belongs to at least one of these progressions?
1992 India National Olympiad, 7
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.
2006 Tournament of Towns, 4
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)
2022 Bolivia Cono Sur TST, P3
Is it possible to complete the following square knowning that each row and column make an aritmetic progression?
1994 Spain Mathematical Olympiad, 1
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.
2004 Tournament Of Towns, 1
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.
2019 Saint Petersburg Mathematical Olympiad, 1
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.
2022/2023 Tournament of Towns, P4
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]
2015 Estonia Team Selection Test, 1
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 $a_1 = a_2 = . . . = a_n$.
2017 Bosnia And Herzegovina - Regional Olympiad, 3
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
2006 Dutch Mathematical Olympiad, 3
$1+2+3+4+5+6=6+7+8$.
What is the smallest number $k$ greater than $6$ for which:
$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?
2016 Regional Olympiad of Mexico Center Zone, 5
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?