This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 93

1978 Romania Team Selection Test, 5
Tags: arithmetic progression , geometry
Prove that there is no square with its four vertices on four concentric circles whose radii form an arithmetic progression.

1983 IMO, 2
Tags: extremal combinatorics , ramsey theory , arithmetic progression , arithmetic sequence , 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?

2014 Albania Round 2, 2
Tags: arithmetic progression , sequence , algebra
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

Fractal Edition 1, P1
Tags: arithmetic progression
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.

2020 Moldova Team Selection Test, 1
Tags: arithmetic progression , geometric progression , progression , number theory
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)

2015 Junior Balkan Team Selection Tests - Romania, 3
Tags: arithmetic progression , partition , arithmetic sequence , ratio , combinatorics
Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

2014 Danube Mathematical Competition, 3
Tags: arithmetic sequence , arithmetic progression , coprime , number theory
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

2020 Estonia Team Selection Test, 1
Tags: arithmetic sequence , arithmetic progression , algebra
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

2020 Nordic, 1
Tags: number theory , arithmetic progression , min , 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]

2019 District Olympiad, 3
Tags: algebra , sequence , arithmetic progression , floor function
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.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.8
Tags: arithmetic progression , algebra
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?

1989 Tournament Of Towns, (240) 4
Tags: algebra , arithmetic progression , inequalities
The set of natural numbers is represented as a union of pairwise disjoint subsets, whose elements form infinite arithmetic progressions with positive differences $d_1,d_2,d_3,...$. Is it possible that the sum $\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... $ does not exceed $0.9$? Consider the cases where (a) the total number of progressions is finite, and (b) the number of progressions is infinite. (In this case the condition that $\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... $ does not exceed $0.9$ should be taken to mean that the sum of any finite number of terms does not exceed 0.9.) (A. Tolpugo, Kiev)

1986 Tournament Of Towns, (127) 2
Tags: combinatorics , arithmetic progression , difference , algebra
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?

1935 Moscow Mathematical Olympiad, 007
Tags: arithmetic progression , system of equations , geometric progression , algebra
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.

2024 Romania National Olympiad, 4
Tags: arithmetic progression , sequence , algebra
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.

1997 Spain Mathematical Olympiad, 1
Tags: arithmetic progression , algebra
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 $1$.

1980 Austrian-Polish Competition, 1
Tags: sequence , algebra , arithmetic progression , arithmetic 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.

IV Soros Olympiad 1997 - 98 (Russia), 9.2
Tags: arithmetic progression , number theory , algebra
The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?

2022/2023 Tournament of Towns, P6
Tags: arithmetic progression , number theory
Let $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$? [i]Viktor Kleptsyn[/i]

1983 Czech and Slovak Olympiad III A, 4
Tags: arithmetic sequence , binomial coefficient , binomial coefficient identity , arithmetic progression , algebra
Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

2007 Mathematics for Its Sake, 2
Tags: arithmetic sequence , sequence , real analysis , limit , arithmetic progression
Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression of positive real numbers, and $ m $ be a natural number. Calculate: [b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [b]b)[/b] $ \lim_{n\to\infty } \frac{1}{a_n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [i]Dumitru Acu[/i]

1980 IMO Longlists, 13
Tags: arithmetic progression , sequence , algebra , arithmetic 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.

1990 Chile National Olympiad, 3
Tags: arithmetic progression , combinatorics
Given a polygon with $n$ sides, we assign the numbers $0,1,...,n-1$ to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for $n = 5$. Notice that the numbers assigned to the sides are still in arithmetic progression. [img]https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png[/img] $\bullet$ Make the respective assignment for a $9$-sided polygon, and generalize for odd $n$. $\bullet$ Prove that this is not possible if $n$ is even.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.1
Tags: algebra , arithmetic progression
The equation $x^2 + bx + c = 0$ has two different roots $x_1$ and $x_2$. It is also known that the numbers $b$, $x_1$, $c$, $x_2$ in the indicated order form an arithmetic progression. Find the difference of this progression.

2021 German National Olympiad, 6
Tags: perfect square , arithmetic progression , number theory
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.