Found problems: 492
2023 Girls in Mathematics Tournament, 1
Define $(a_n)$ a sequence, where $a_1= 12, a_2= 24$ and for $n\geq 3$, we have: $$a_n= a_{n-2}+14$$
a) Is $2023$ in the sequence?
b) Show that there are no perfect squares in the sequence.
2012 Iran MO (3rd Round), 2
Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that
$W(k,2)=\Omega (2^{\frac{k}{2}})$.
PEN A Problems, 2
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.
2024 New Zealand MO, 2
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]$x$.
2004 Romania National Olympiad, 1
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$.
Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.
2019 IMO Shortlist, A7
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying
\[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\]
for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set
\[X_v=\{x\in\mathbb Z:f(x)=v\}\]
is finite and nonempty.
(a) Prove that there exists such a function $f$ for which there is an $f$-rare integer.
(b) Prove that no such function $f$ can have more than one $f$-rare integer.
[i]Netherlands[/i]
2020 Regional Olympiad of Mexico West, 5
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 $$x_4-x_3=x_3-x_2=x_2-x_1$$
1990 Irish Math Olympiad, 2
Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.
1991 IMO Shortlist, 16
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
1967 IMO Longlists, 33
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.
2005 USAMTS Problems, 1
An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proof how many of the 36 possible sequences formed in this way contain at least one perfect square.
1955 AMC 12/AHSME, 32
If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that:
$ \textbf{(A)}\ \text{they form an arithmetic progression} \\
\textbf{(B)}\ \text{they form a geometric progression} \\
\textbf{(C)}\ \text{they are unequal} \\
\textbf{(D)}\ \text{they are all negative numbers} \\
\textbf{(E)}\ \text{only b is negative and a and c are positive}$
1970 IMO Longlists, 53
A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.
1994 AMC 12/AHSME, 20
Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$
2010 Singapore MO Open, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
2006 IberoAmerican Olympiad For University Students, 2
Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation
\[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\]
has at least one real root.
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]
2010 Swedish Mathematical Competition, 4
We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$.
Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.
2004 AIME Problems, 9
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 $n+a_n$.
1997 Romania Team Selection Test, 3
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.
PEN K Problems, 5
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
2022 AMC 10, 20
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence?
$\textbf{(A) }190\qquad\textbf{(B) }194\qquad\textbf{(C) }198\qquad\textbf{(D) }202\qquad\textbf{(E) }206$
2015 Ukraine Team Selection Test, 12
For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.
1997 AMC 12/AHSME, 20
Which one of the following integers can be expressed as the sum of $ 100$ consecutive positive integers?
$ \textbf{(A)}\ 1,\!627,\!384,\!950\qquad \textbf{(B)}\ 2,\!345,\!678,\!910\qquad \textbf{(C)}\ 3,\!579,\!111,\!300\qquad \textbf{(D)}\ 4,\!692,\!581,\!470\qquad \textbf{(E)}\ 5,\!815,\!937,\!260$
PEN O Problems, 35
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 \}$.