Olympiad problems

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.

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

2002 Estonia National Olympiad, 1

Tags: arithmetic progression , algebra , polynomial , arithmetic sequence

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

2017 Israel Oral Olympiad, 7

Tags: combinatorics , arithmetic progression , arithmetic sequence

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 $N$?

2005 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , arithmetic progression , divisible , divide

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 $2005$?

2000 Estonia National Olympiad, 1

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

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.

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.

2021 German National Olympiad, 6

Tags: arithmetic progression , perfect square , 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.

2020 Iranian Our MO, 5

Tags: arithmetic progression , number theory

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]

IV Soros Olympiad 1997 - 98 (Russia), 10.3

Tags: number theory , arithmetic progression , digit

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?

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

1959 Polish MO Finals, 6

Tags: geometry , arithmetic progression , arithmetic sequence

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.

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

2023 OMpD, 2

Tags: floor function , algebra , bounding , arithmetic progression

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

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.

1975 Vietnam National Olympiad, 4

Tags: digit , arithmetic progression , number theory

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

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.

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.

2014 Albania Round 2, 2

Tags: algebra , arithmetic progression , sequence

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

1980 IMO, 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.

2018 Auckland Mathematical Olympiad, 4

Tags: combinatorics , game , winning strategy , arithmetic progression

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

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.

2004 Tournament Of Towns, 4

Tags: integer , arithmetic progression , sequence , 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.

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]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.8

Tags: algebra , arithmetic progression

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?