Olympiad problems

1992 India National Olympiad, 7

Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

1991 Tournament Of Towns, (319) 6

Tags: number theory , arithmetic progression , digit

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)

2019 IFYM, Sozopol, 1

Tags: algebra , arithmetic progression

Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers.

2007 Mathematics for Its Sake, 2

Tags: limit , arithmetic sequence , real analysis , arithmetic progression , sequence

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]

2021 Irish Math Olympiad, 6

Tags: geometry , equilateral , algebra , arithmetic progression

A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.

2006 Tournament of Towns, 4

Tags: geometric progression , arithmetic progression , sequence , integer , number theory

Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)

2017 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: arithmetic progression , number theory , set

Let $S$ be a set of $6$ positive real numbers such that $\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S$ or $a-b \in S$ Prove that if we sort these numbers in ascending order, then they form an arithmetic progression

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.

2020 Estonia Team Selection Test, 1

Tags: arithmetic sequence , algebra , arithmetic progression

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

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?

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)

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?

1980 Austrian-Polish Competition, 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.

2008 Bulgarian Autumn Math Competition, Problem 11.1

Tags: arithmetic progression , algebra

Let $a_{1},a_{2},\ldots$ be an infinite arithmetic progression. It's known that there exist positive integers $p,q,t$ such that $a_{p}+tp=a_{q}+tq$. If $a_{t}=t$ and the sum of the first $t$ numbers in the sequence is $18$, determine $a_{2008}$.

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.

2022/2023 Tournament of Towns, P4

Tags: number theory , arithmetic progression

Let $a_1, a_2, a_3,\ldots$ and $b_1, b_2, b_3,\ldots$ be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio $a_k/b_k$ is an integer for all k. Is it true that this ratio does not depend on $k{}$? [i]Boris Frenkin[/i]

2004 Argentina National Olympiad, 6

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

2023 Stars of Mathematics, 4

Tags: arithmetic progression , algebra

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.