Olympiad problems

2024 Germany Team Selection Test, 3

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

2010 Romania Team Selection Test, 3

Tags: arithmetic sequence , number theory , modular arithmetic

Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.) [i]Vlad Matei[/i]

2018 Thailand TST, 3

Tags: number theory , arithmetic sequence

Does there exist an arithmetic progression with $2017$ terms such that each term is not a perfect power, but the product of all $2017$ terms is?

2021 Romanian Master of Mathematics Shortlist, C1

Tags: combinatorics , algebra , graph theory , arithmetic progression , arithmetic sequence

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.

2000 Estonia National Olympiad, 1

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

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.

2007 AMC 12/AHSME, 24

Tags: function , algebra , trigonometry , floor function , domain , arithmetic sequence , modular arithmetic

For each integer $ n > 1,$ let $ F(n)$ be the number of solutions of the equation $ \sin x \equal{} \sin nx$ on the interval $ [0,\pi].$ What is $ \sum_{n \equal{} 2}^{2007}F(n)?$ $ \textbf{(A)}\ 2,014,524 \qquad \textbf{(B)}\ 2,015,028 \qquad \textbf{(C)}\ 2,015,033 \qquad \textbf{(D)}\ 2,016,532 \qquad \textbf{(E)}\ 2,017,033$

2019 Saudi Arabia Pre-TST + Training Tests, 1.2

Tags: algebra , arithmetic sequence , polynomial

Let $P(x)$ be a polynomial of degree $n \ge 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.

2004 India IMO Training Camp, 4

Tags: function , arithmetic sequence , number theory

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

2003 AIME Problems, 8

Tags: modular arithmetic , arithmetic sequence , geometric sequence , ratio

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.

1995 Tournament Of Towns, (482) 6

Tags: number theory , arithmetic sequence , sum of digits

Does there exist an increasing arithmetic progression of (a) $11$ (b) $10000$ (c) infinitely many positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression? (A Shapovalov)

2000 Belarus Team Selection Test, 4.3

Tags: algebra , arithmetic sequence , divisibility , sum of digits , modular arithmetic

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

2012 Federal Competition For Advanced Students, Part 2, 1

Tags: algebra , arithmetic sequence , induction

Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$. If there is a real number $C$, such that \[ (i-j)m_k+(j-k)m_i+(k-i)m_j=C\] for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.

2013 Online Math Open Problems, 4

Tags: arithmetic sequence

Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Proposed by Evan Chen[/i]

2010 Singapore MO Open, 3

Tags: number theory , prime number , arithmetic sequence , ratio

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

1960 Putnam, B4

Tags: arithmetic sequence , power

Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.

2001 CentroAmerican, 2

Tags: arithmetic sequence , function , quadratic , absolute value

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

1998 Romania Team Selection Test, 2

Tags: arithmetic sequence

Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that \[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\] where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.