Olympiad problems

1998 VJIMC, Problem 1

Let $a$ and $d$ be two positive integers. Prove that there exists a constant $K$ such that every set of $K$ consecutive elements of the arithmetic progression $\{a+nd\}_{n=1}^\infty$ contains at least one number which is not prime.

1975 Vietnam National Olympiad, 4

Tags: Arithmetic Progression , Digit , number theory

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

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.

1998 China Team Selection Test, 1

Tags: arithmetic sequence , combinatorics unsolved , combinatorics , arithmetic , Arithmetic Progression , China , TST

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

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 $n$.

2020 Kazakhstan National Olympiad, 4

Tags: Arithmetic Progression , game strategy , combinatorics , arithmetic sequence

Alice and Bob play a game on the infinite side of a checkered strip, in which the cells are numbered with consecutive integers from left to right (..., $-2$, $-1$, $0$, $1$, $2$, ...). Alice in her turn puts one cross in any free cell, and Bob in his turn puts zeros in any 2020 free cells. Alice will win if he manages to get such 4 cells marked with crosses, the corresponding cell numbers will form an arithmetic progression. Bob's goal in this game is to prevent Alice from winning. They take turns and Alice moves first. Will Alice be able to win no matter how Bob plays?

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 , combinatorics open , 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 $N$?

1998 All-Russian Olympiad Regional Round, 9.1

Tags: geometry , Arithmetic Progression

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.

2007 Mathematics for Its Sake, 2

Tags: arithmetic sequence , real analysis , limits , Sequences , 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]

1990 Chile National Olympiad, 3

Tags: combinatorics , Arithmetic Progression

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.

2018 Puerto Rico Team Selection Test, 1

Tags: Arithmetic Progression , algebra

Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?

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 $ .

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?

IV Soros Olympiad 1997 - 98 (Russia), 9.2

Tags: number theory , algebra , Arithmetic Progression

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?

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: Arithmetic Progression , partitions , 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 ?

2004 Tournament Of Towns, 4

Tags: Integers , 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.

1983 IMO Longlists, 50

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?

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.

1983 Czech and Slovak Olympiad III A, 4

Tags: algebra , Binomial coefficient identity , Arithmetic Progression , national olympiad , arithmetic sequence , binomial coefficients

Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

1998 China Team Selection Test, 1

Tags: arithmetic sequence , combinatorics unsolved , combinatorics , arithmetic , Arithmetic Progression , China , TST

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

2009 Tournament Of Towns, 4

Tags: geometric progression , Arithmetic Progression , algebra

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?

2020 Moldova Team Selection Test, 1

Tags: progressions , Arithmetic Progression , geometric 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)

Estonia Open Senior - geometry, 2002.1.2

Tags: geometry , right triangle , Arithmetic Progression , inradius

The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.