Olympiad problems

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 $81$.

2020 Estonia Team Selection Test, 1

Tags: arithmetic sequence , algebra , arithmetic progression

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

1983 IMO Longlists, 50

Tags: arithmetic sequence , extremal combinatorics , ramsey theory , arithmetic progression , combinatorics

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?

1992 India National Olympiad, 7

Tags: combinatorics , counting , chessboard , arithmetic progression

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.

1955 Moscow Mathematical Olympiad, 313

Tags: combinatorics , combinatorial geometry , equal segments , arithmetic progression

On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.

1997 Spain Mathematical Olympiad, 1

Tags: algebra , arithmetic progression

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 $1$.

2022 Kyiv City MO Round 1, Problem 4

Tags: algebra , arithmetic progression , arithmetic sequence

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]

2015 Silk Road, 2

Tags: perfect square , arithmetic progression , number theory , sequence

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 $n$ .

2024 Romania National Olympiad, 4

Tags: algebra , arithmetic progression , sequence

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.

2021 Romanian Master of Mathematics Shortlist, C1

Tags: graph theory , arithmetic progression , combinatorics , arithmetic sequence , algebra

Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.

2017 Singapore MO Open, 4

Tags: algebra , inequalities , arithmetic progression , geometric progression , sequence

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

1978 Romania Team Selection Test, 5

Tags: geometry , arithmetic progression

Prove that there is no square with its four vertices on four concentric circles whose radii form an arithmetic progression.

2016 Costa Rica - Final Round, LR3

Tags: algebra , arithmetic progression , arithmetic sequence

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.

2008 Bulgarian Autumn Math Competition, Problem 11.1

Tags: arithmetic progression , algebra

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 Nordic, 1

Tags: number theory , min , arithmetic progression , prime

For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]

1923 Eotvos Mathematical Competition, 3

Tags: number theory , arithmetic progression

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

2015 Estonia Team Selection Test, 1

Tags: algebra , consecutive , arithmetic progression , arithmetic sequence

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 $a_1 = a_2 = . . . = a_n$.

1980 Austrian-Polish Competition, 1

Tags: algebra , arithmetic sequence , arithmetic progression , sequence

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.

Kvant 2020, M2627

Tags: algebra , arithmetic progression

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]

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: arithmetic progression , arithmetic sequence , ratio , combinatorics , partition

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

2020 Estonia Team Selection Test, 1

Tags: arithmetic sequence , algebra , arithmetic progression

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

1974 Chisinau City MO, 77

Tags: arithmetic progression , algebra

Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?

2004 Argentina National Olympiad, 6

Tags: arithmetic progression , algebra , arithmetic sequence

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 $2b=a+c$.

1935 Moscow Mathematical Olympiad, 008

Tags: arithmetic progression , inradius , altitude , geometry , arithmetic sequence

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.

2019 District Olympiad, 3

Tags: sequence , floor function , arithmetic progression , algebra

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$ $\textbf{a)}$ Prove that the given sequence is an arithmetic progression. $\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.