Found problems: 93
1980 Austrian-Polish Competition, 1
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.
2017 Singapore Senior Math Olympiad, 5
Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.
2000 Estonia National Olympiad, 1
Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence.
Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.
2015 NZMOC Camp Selection Problems, 5
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 $d$?
1959 Polish MO Finals, 6
Given a triangle in which the sides $ a $, $ b $, $ c $ form an arithmetic progression and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle.
1980 IMO, 1
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.
1997 Spain Mathematical Olympiad, 1
Compute the sum of the squares of the first $100$ terms of an arithmetic progression, given that their sum is $-1$ and that the sum of those among them having an even index is $1$.
2014 Albania Round 2, 2
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
2019 IFYM, Sozopol, 1
Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers.
1962 Polish MO Finals, 1
Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then
$$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$
2014 Danube Mathematical Competition, 3
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
2008 Bulgarian Autumn Math Competition, Problem 11.1
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 $a_{2008}$.
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
2005 Mexico National Olympiad, 4
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.
1923 Eotvos Mathematical Competition, 3
Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.
2010 Peru MO (ONEM), 2
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 $81$.
1935 Moscow Mathematical Olympiad, 008
Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.
2020 Iranian Our MO, 5
Concider two sequences $x_n=an+b$, $y_n=cn+d$ where $a,b,c,d$ are natural numbers and $gcd(a,b)=gcd(c,d)=1$, prove that there exist infinite $n$ such that $x_n$, $y_n$ are both square-free.
[i]Proposed by Siavash Rahimi Shateranloo, Matin Yadollahi[/i] [b]Rated 3[/b]
1983 IMO Shortlist, 14
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?
2004 Argentina National Olympiad, 6
Decide if it is possible to generate an infinite sequence of positive integers $a_n$ such that in the sequence there are no three terms that are in arithmetic progression and that for all $n$ $\left |a_n-n^2\right | <\frac{n}{2}$.
Clarification: Three numbers $a$, $b$, $c$ are in arithmetic progression if and only if $2b=a+c$.
Kvant 2020, M2627
An infinite arithmetic progression is given. The products of the pairs of its members are considered. Prove that two of these numbers differ by no more than 1.
[i]Proposed by A. Kuznetsov[/i]
Albania Round 2, 2
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
2021 Irish Math Olympiad, 6
A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.
2016 Costa Rica - Final Round, LR3
Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression.
2002 Argentina National Olympiad, 6
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.