Olympiad problems

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]

1980 IMO Shortlist, 13

Tags: sequence , arithmetic progression , arithmetic sequence , algebra

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.

2004 Tournament Of Towns, 4

Tags: sequence , integer , arithmetic progression , number theory

Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.

2017 Israel Oral Olympiad, 7

Tags: arithmetic sequence , combinatorics , arithmetic progression

The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?

1955 Polish MO Finals, 2

Tags: number theory , arithmetic progression

Prove that among the seven natural numbers forming an arithmetic progression with difference $ 30 $ , one and only one is divisible by $ 7 $ .

2015 NZMOC Camp Selection Problems, 5

Tags: algebra , arithmetic sequence , arithmetic progression

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 $d$?

1998 All-Russian Olympiad Regional Round, 9.1

Tags: arithmetic progression , geometry

The lengths of the sides of a certain triangle and the diameter of the inscribed part circles are four consecutive terms of arithmetic progression. Find all such triangles.

1983 IMO Longlists, 50

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?

2011 IFYM, Sozopol, 4

Tags: set , arithmetic progression , arithmetic sequence , number theory

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.

1975 Vietnam National Olympiad, 4

Tags: digit , number theory , arithmetic progression

Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.

1959 Polish MO Finals, 6

Tags: arithmetic sequence , arithmetic progression , geometry

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.

1962 Polish MO Finals, 1

Tags: algebra , arithmetic progression , sum

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 $$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$

1994 Spain Mathematical Olympiad, 1

Tags: number theory , arithmetic progression , perfect square

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

1964 Poland - Second Round, 3

Tags: arithmetic progression , arithmetic sequence , divisible , number theory

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

Tags: algebra , real root , arithmetic progression , arithmetic sequence , polynomial

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression

1974 Chisinau City MO, 77

Tags: algebra , arithmetic progression

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?

2017 Singapore Senior Math Olympiad, 5

Tags: algebra , number theory , combinatorics , arithmetic progression , sequence , integer

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.

Oliforum Contest I 2008, 1

Tags: arithmetic sequence , arithmetic progression , number theory

(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.

2002 Estonia National Olympiad, 1

Tags: arithmetic progression , arithmetic sequence , polynomial , algebra

Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.

2023 OMpD, 2

Tags: bounding , arithmetic progression , floor function , algebra

Find all pairs $(a,b)$ of real numbers such that $\lfloor an + b \rfloor$ is a perfect square, for all positive integer $n$.

2004 Argentina National Olympiad, 6

Tags: algebra , arithmetic progression , 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 $2b=a+c$.

2009 Tournament Of Towns, 4

Tags: algebra , geometric progression , arithmetic progression

Consider an infinite sequence consisting of distinct positive integers such that each term (except the rst one) is either an arithmetic mean or a geometric mean of two neighboring terms. Does it necessarily imply that starting at some point the sequence becomes either arithmetic progression or a geometric progression?

2006 Dutch Mathematical Olympiad, 3

Tags: arithmetic progression , algebra , sum , minimum

$1+2+3+4+5+6=6+7+8$. What is the smallest number $k$ greater than $6$ for which: $1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?

2015 Estonia Team Selection Test, 1

Tags: consecutive , arithmetic progression , arithmetic sequence , algebra

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

2004 Spain Mathematical Olympiad, Problem 1

Tags: matrix , matrices , arithmetic progression , algebra

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 ${2004}$.