Found problems: 492
1988 IberoAmerican, 1
The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.
1952 AMC 12/AHSME, 30
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:
$ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 1 \qquad\textbf{(C)}\ 1: 4 \qquad\textbf{(D)}\ 4: 1 \qquad\textbf{(E)}\ 1: 1$
1998 Tuymaada Olympiad, 7
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
2025 Canada Junior National Olympiad, 1
Suppose an infinite non-constant arithmetic progression of integers contains $1$ in it. Prove that there are an infinite number of perfect cubes in this progression. (A [i]perfect cube[/i] is an integer of the form $k^3$, where $k$ is an integer. For example, $-8$, $0$ and $1$ are perfect cubes.)
2006 Czech-Polish-Slovak Match, 2
There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?
1994 China Team Selection Test, 2
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
2021 Olympic Revenge, 5
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.
1998 National Olympiad First Round, 8
$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ?
$\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$
PEN G Problems, 18
Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.
2011 ISI B.Stat Entrance Exam, 7
[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$.
[b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.
2014 Contests, 1
Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.
1978 IMO Longlists, 10
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.
2016 Argentina National Olympiad, 1
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 $2$.
1998 Singapore Team Selection Test, 3
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.
2024 Moldova Team Selection Test, 8
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.
2011 IFYM, Sozopol, 4
Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.
1995 National High School Mathematics League, 1
In arithmetic sequence $(a_n)$, $3a_8=5a_{13},a_1>0$. Define $S_n=\sum_{i=1}^n a_i$, then the largest number in $(S_n)$ is
$\text{(A)}S_{10}\qquad\text{(B)}S_{11}\qquad\text{(C)}S_{20}\qquad\text{(D)}S_{21}$
1999 Romania Team Selection Test, 7
Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that
\[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \]
Give an example of two such progressions having at least five terms.
[i]Mihai Baluna[/i]
PEN O Problems, 56
Show that it is possible to color the set of integers \[M=\{ 1, 2, 3, \cdots, 1987 \},\] using four colors, so that no arithmetic progression with $10$ terms has all its members the same color.
1997 Argentina National Olympiad, 6
Decide if there are ten natural and distinct numbers $a_1,a_2,\ldots ,a_{10}$ such that:
$\bullet$ Each of them is a power of a natural number with a natural exponent and greater than $1$.
$\bullet$ The numbers $a_1,a_2,\ldots ,a_{10}$ form an arithmetic progression.
2015 Romania Team Selection Tests, 3
Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .
2009 APMO, 4
Prove that for any positive integer $ k$, there exists an arithmetic sequence $ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}$ of rational numbers, where $ a_i, b_i$ are relatively prime positive integers for each $ i \equal{} 1,2,...,k$ such that the positive integers $ a_1, b_1, a_2, b_2, ..., a_k, b_k$ are all distinct.
2006 Grigore Moisil Urziceni, 2
Let be an infinite sequence $ \left( c_n \right)_{n\ge 1} $ of positive real numbers, with $ c_1=1, $ and satisfying
$$ c_{n+1}-\frac{1}{c_{n+1}} =c_n+\frac{1}{c_n} , $$
for all natural numbers $ n. $ Prove that:
[b]a)[/b] there exists a natural number $ k $ such that the sequence $ \left( c_n^k+\frac{1}{c_n^k} \right)_{n\ge 1} $ is an arithmetic one.
[b]b)[/b] there exist two sequences $ \left( u_n \right)_{n\ge 1} ,\left( v_n \right)_{n\ge 1} $ of nonegative integers such that $ c_n=\sqrt{u_n} +\sqrt{v_n} , $ for any natural number $ n. $
2008 Iran MO (3rd Round), 2
Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]
2013 AMC 10, 19
The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root?
$\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $