Olympiad problems

1955 Moscow Mathematical Olympiad, 313

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.

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.

2021 Romanian Master of Mathematics Shortlist, C1

Tags: graph theory , Arithmetic Progression , RMM Shortlist , 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.

1983 IMO, 2

Tags: arithmetic sequence , Extremal combinatorics , Ramsey Theory , Arithmetic Progression , IMO , IMO 1983 , 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?

2004 Spain Mathematical Olympiad, Problem 1

Tags: math olympiad , Matrices , matrix , Arithmetic Progression , algebra unsolved

We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.

2015 Silk Road, 2

Tags: Perfect Square , Arithmetic Progression , number theory , Sequences , 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 $n$ .

2005 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , Arithmetic Progression , divisible , divides

Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

1955 Moscow Mathematical Olympiad, 299

Tags: number theory , Arithmetic Progression , Sequence , primes , divisible

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

2020 Nordic, 1

Tags: number theory , min , Arithmetic Progression , primes

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 Malaysia National Olympiad, B3

Tags: arithmetic sequence , Arithmetic Progression , algebra

An arithmetic sequence of five terms is considered $good$ if it contains 19 and 20. For example, $18.5,19.0,19.5,20.0,20.5$ is a $good$ sequence. For every $good$ sequence, the sum of its terms is totalled. What is the total sum of all $good$ sequences?

IV Soros Olympiad 1997 - 98 (Russia), 10.3

Tags: number theory , Arithmetic Progression , Digits

Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?

2021 German National Olympiad, 6

Tags: number theory , number theory proposed , Arithmetic Progression , Perfect Squares , Perfect Square

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.

Oliforum Contest I 2008, 1

Tags: number theory , Arithmetic Progression , arithmetic sequence

(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression. (b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.

2023 Stars of Mathematics, 4

Tags: algebra , Arithmetic Progression

Determine all integers $n\geqslant 3$ such that there exist $n{}$ pairwise distinct real numbers $a_1,\ldots,a_n$ for which the sums $a_i+a_j$ over all $1\leqslant i<j\leqslant n$ form an arithmetic progression.

1935 Moscow Mathematical Olympiad, 007

Tags: Arithmetic Progression , algebra , system of equations , geometric progression

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 $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.

1989 Tournament Of Towns, (240) 4

Tags: algebra , inequalities , Arithmetic Progression

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)

1991 Tournament Of Towns, (319) 6

Tags: number theory , Arithmetic Progression , Digits

An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than $100$. (b) Give an example of such a progression with $72$ terms. (c) Prove that the number of terms in any such progression does not exceed $72$. (V. Bugaenko and Tarasov, Moscow)

2022/2023 Tournament of Towns, P6

Tags: number theory , Arithmetic Progression

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]