Found problems: 492
2015 Ukraine Team Selection Test, 7
Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements
1992 Baltic Way, 3
Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).
1992 IMTS, 4
In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers, \[ 113,137,149,155,173, \] accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?
1987 IMO Longlists, 53
Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.
[b][i]Alternative formulation[/i][/b]
Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.
[i]Proposed by Romania[/i]
2017 Purple Comet Problems, 1
Paul starts at $1$ and counts by threes: $1, 4, 7, 10, ... $. At the same time and at the same speed, Penny counts backwards from $2017$ by fi ves: $2017, 2012, 2007, 2002,...$ . Find the one number that both Paul and Penny count at the same time.
2017 Philippine MO, 3
Each of the numbers in the set \(A = \{1,2, \cdots, 2017\}\) is colored either red or white. Prove that for \(n \geq 18\), there exists a coloring of the numbers in \(A\) such that any of its n-term arithmetic sequences contains both colors.
1967 IMO Shortlist, 4
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.
2015 China Western Mathematical Olympiad, 6
For a sequence $a_1,a_2,...,a_m$ of real numbers, define the following sets
\[A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}\]
Let $n$ be a given integer, and $n>2$. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set $A\triangle B$, where $A\triangle B$ $= \left(A\cup B\right) \setminus \left(A\cap B\right).$
1973 AMC 12/AHSME, 28
If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence
$ \textbf{(A)}\ \text{which is a G.P} \qquad$
$ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$
$ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$
$ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$
$ \textbf{(E)}\ \text{none of these}$
2014 Danube Mathematical Competition, 3
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
2012 Vietnam National Olympiad, 2
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
2010 Dutch IMO TST, 2
Let $A$ and $B$ be positive integers. De fine the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.
2000 AMC 12/AHSME, 14
When the mean, median, and mode of the list
\[ 10, 2, 5, 2, 4, 2, x\]are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $ x$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 20$
2015 Benelux, 4
Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain?
(An [i]arithmetic progression[/i] is a set of the form $\{a,a+d,\dots,a+kd\}$, where $a,d,k$ are positive integers, and $k\geqslant 2$; thus an arithmetic progression has at least three elements, and successive elements have difference $d$, called the [i]common difference[/i] of the arithmetic progression.)
1999 National Olympiad First Round, 8
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
1988 AMC 8, 19
What is the $100th$ number in the arithmetic sequence: $ 1,5,9,13,17,21,25,... $
$ \text{(A)}\ 397\qquad\text{(B)}\ 399\qquad\text{(C)}\ 401\qquad\text{(D)}\ 403\qquad\text{(E)}\ 405 $
2019 India PRMO, 20
Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.
2004 AMC 10, 18
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 of the geometric progression?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 36\qquad
\textbf{(D)}\ 49\qquad
\textbf{(E)}\ 81$
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
2004 China Western Mathematical Olympiad, 1
The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$,
\[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \]
Find $a_{2004}$.
Oliforum Contest I 2008, 1
(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.
(b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.
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.
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2008 ITest, 43
Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teahcers at school taught her about how a young Karl Gauss quickly computed the sum \[1+2+3+\cdots+98+99+100\] in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?
2002 India IMO Training Camp, 17
Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.