Olympiad problems

1987 AMC 12/AHSME, 9

The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $

PEN K Problems, 5

Tags: functional equation , function , arithmetic sequence

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

2024 Moldova Team Selection Test, 8

Tags: arithmetic sequence , number theory

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.

2024 New Zealand MO, 2

Tags: arithmetic sequence , algebra

Consider the sequence $a_{1}, a_{2}, a_{3},\ldots$ defined by $a_{1}=2024^{2024}$ and for each positive integer $n$, $$a_{n+1}=\left|a_{n}-\sqrt{2}\right|.$$ Prove that there exists an integer $k$ such that $a_{k+2}=a_k$. [i]Here [/i]$\left|x\right|$[i] denotes the absolute value of [/i]$x$.

2002 AMC 12/AHSME, 5

Tags: arithmetic sequence

Let $ v$, $ w$, $ x$, $ y$, and $ z$ be the degree measures of the five angles of a pentagon. Suppose $ v < w < x < y < z$ and $ v$, $ w$, $ x$, $ y$, and $ z$ form an arithmetic sequence. Find the value of $ x$. $ \textbf{(A)}\ 72 \qquad \textbf{(B)}\ 84 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 120$

2020 Regional Olympiad of Mexico West, 5

Tags: arithmetic sequence , algebra

Determine the values that $n$ can take so that the equation in $ x $ $$ x^4-(3n+2)x^2+n^2=0$$ has four different real roots $ x_1$, $x_2$, $x_3$ and $x_4$ in arithmetic progression. That is, they satisfy that $$x_4-x_3=x_3-x_2=x_2-x_1$$

2024 Thailand TST, 1

Tags: number theory , arithmetic sequence

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.

2019 Saudi Arabia Pre-TST + Training Tests, 1.2

Tags: divide , arithmetic sequence , number theory

Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

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.

1994 AIME Problems, 1

Tags: modular arithmetic , arithmetic sequence , number theory

The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?

1969 IMO Longlists, 17

Tags: equation , arithmetic sequence , number theory , algebra

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

2011 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , arithmetic sequence , system of equations , number theory , greatest common divisor , relatively prime , algebra

We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality \[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\] holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i]. [b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

2021 Kyiv Mathematical Festival, 1

Tags: combinatorial geometry , arithmetic sequence , geometry

Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)

2019 Centers of Excellency of Suceava, 3

Tags: bijection , geometric sequence , permutation of the naturals , arithmetic sequence , algebra

Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

PEN N Problems, 13

Tags: arithmetic sequence , more sequences , polynomial , algebra

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

2012 Math Prize For Girls Problems, 11

Tags: geometric sequence , arithmetic sequence

Alison has an analog clock whose hands have the following lengths: $a$ inches (the hour hand), $b$ inches (the minute hand), and $c$ inches (the second hand), with $a < b < c$. The numbers $a$, $b$, and $c$ are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day: $A$ inches (the hour hand), $B$ inches (the minute hand), and $C$ inches (the second hand). The numbers $A$, $B$, and $C$ (in this order) are consecutive terms of a geometric sequence. What is the value of $\frac{B}{A}$?

2008 Iran MO (3rd Round), 2

Tags: number theory , arithmetic sequence

Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]

2006 Grigore Moisil Urziceni, 3

Tags: integer sequence , numerical analysis , sequence , number theory , arithmetic sequence

Let be a sequence $ \left( b_n \right)_{n\ge 1} $ of integers, having the following properties: $ \text{(i)} $ the sequence $ \left( \frac{b_n}{n} \right)_{n\ge 1} $ is convergent. $ \text{(ii)} m-n|b_m-b_n, $ for any natural numbers $ m>n. $ Prove that there exists an index from which the sequence $ \left( b_n \right)_{n\ge 1} $ is an arithmetic one. [i]Cristinel Mortici[/i]

1994 Bundeswettbewerb Mathematik, 4

Tags: arithmetic sequence , geometric sequence , rational , algebra

Let $a,b$ be real numbers ($b\ne 0$) and consider the infinite arithmetic sequence $a, a+b ,a +2b , \ldots.$ Show that this sequence contains an infinite geometric subsequence if and only if $\frac{a}{b}$ is rational.

2008 China Team Selection Test, 3

Tags: combinatorics , ramsey theory , arithmetic sequence , search

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

1997 IMO Shortlist, 15

Tags: perfect cube , perfect square , arithmetic sequence , number theory , modular arithmetic

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

2016 Argentina National Olympiad, 1

Tags: number theory , arithmetic sequence , perfect power

Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power. Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.

2003 IMO Shortlist, 4

Tags: arithmetic sequence , inequalities

Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

2010 AMC 12/AHSME, 10

Tags: algebra , arithmetic sequence , system of equations

The first four terms of an arithmetic sequence are $ p,9,3p\minus{}q,$ and $ 3p\plus{}q$. What is the $ 2010^{\text{th}}$ term of the sequence? $ \textbf{(A)}\ 8041\qquad \textbf{(B)}\ 8043\qquad \textbf{(C)}\ 8045\qquad \textbf{(D)}\ 8047\qquad \textbf{(E)}\ 8049$

2010 USAJMO, 2

Tags: pigeonhole principle , arithmetic sequence , induction

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.