Olympiad problems

1999 APMO, 1

Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.

2004 India IMO Training Camp, 4

Tags: arithmetic sequence , number theory , function

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 $f(a) < f(a+d) <f(a+2d)$

1992 IMO Longlists, 46

Tags: number theory , power of 2 , arithmetic sequence

Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$

2009 Romania National Olympiad, 4

Tags: arithmetic sequence , number theory , ratio

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

1981 AMC 12/AHSME, 11

Tags: arithmetic sequence , integration , calculus

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length $\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$

2013 Romania National Olympiad, 1

Tags: arithmetic sequence , induction , algebra

A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.

2014 Postal Coaching, 5

Tags: arithmetic sequence , polynomial , number theory , modular arithmetic , algebra

Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.

2006 Singapore MO Open, 2

Tags: algebra , arithmetic sequence

Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\]

2004 AIME Problems, 9

Tags: arithmetic sequence , geometric sequence

A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1$, the terms $a_{2n-1}$, $a_{2n}$, $a_{2n+1}$ are in geometric progression, and the terms $a_{2n}$, $a_{2n+1}$, and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000. Find $n+a_n$.

2016 HMIC, 5

Tags: additive combinatorics , hmmt , arithmetic sequence , number theory , combinatorics

Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning \[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \] Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$. [i]Yang Liu[/i]

1980 Austrian-Polish Competition, 1

Tags: sequence , algebra , arithmetic progression , arithmetic 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.

2011 All-Russian Olympiad, 2

Tags: arithmetic sequence , quadratic , vieta , invariant , polynomial , induction , algebra

In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?

PEN O Problems, 14

Tags: arithmetic sequence

Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.

1965 AMC 12/AHSME, 20

Tags: arithmetic sequence

For every $ n$ the sum of $ n$ terms of an arithmetic progression is $ 2n \plus{} 3n^2$. The $ r$th term is: $ \textbf{(A)}\ 3r^2 \qquad \textbf{(B)}\ 3r^2 \plus{} 2r \qquad \textbf{(C)}\ 6r \minus{} 1 \qquad \textbf{(D)}\ 5r \plus{} 5 \qquad \textbf{(E)}\ 6r \plus{} 2 \qquad$

1987 Vietnam National Olympiad, 1

Tags: arithmetic sequence , trigonometry , algebra

Let $ u_1$, $ u_2$, $ \ldots$, $ u_{1987}$ be an arithmetic progression with $ u_1 \equal{} \frac {\pi}{1987}$ and the common difference $ \frac {\pi}{3974}$. Evaluate \[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right) \]

2004 AMC 12/AHSME, 14

Tags: quadratic , arithmetic sequence , geometric sequence

A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 81$

2012 Vietnam Team Selection Test, 3

Tags: geometry , arithmetic sequence , combinatorial geometry , symmetry , geometric transformation , modular arithmetic , floor function

Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition: For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.

1967 IMO Longlists, 33

Tags: system of equations , matrix , linear algebra , arithmetic sequence , algebra

In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

2018 Saudi Arabia IMO TST, 2

Tags: inequalities , arithmetic sequence , algebra , geometric sequence

a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$. b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.

2012 NIMO Summer Contest, 4

Tags: arithmetic sequence

The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$. [i]Proposed by Lewis Chen[/i]

2013 Brazil Team Selection Test, 3

Tags: arithmetic sequence , algebra , floor function

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2002 India National Olympiad, 5

Tags: arithmetic sequence , algebra

Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).

2005 All-Russian Olympiad Regional Round, 9.7

Tags: arithmetic sequence , number theory

9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$? ([i]V. Senderov[/i])

2005 Czech And Slovak Olympiad III A, 1

Tags: arithmetic sequence , algebra , sequence , maximum , recurrence relation

Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.

1984 AIME Problems, 1

Tags: arithmetic sequence

Find the value of $a_2 + a_4 + a_6 + \dots + a_{98}$ if $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic progression with common difference 1, and $a_1 + a_2 + a_3 + \dots + a_{98} = 137$.