Found problems: 492
An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.
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 $A\triangle B$ $= \left(A\cup B\right) \setminus \left(A\cap B\right).$
Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a [i]ghost[/i] if it doesn’t appear in $S$.
What is the smallest ghost that is not a multiple of $p$?
[i]Proposed by Guerrero[/i]
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.
[i]AMM Magazine[/i]
Given that $9\times10\times11\times\cdots\times15=32432400$, what is $1\times3\times5\times\cdots\times15$?
[i]2017 CCA Math Bonanza Team Round #1[/i]
A pupil is writing on a board positive integers $x_0,x_1,x_2,x_3...$ after the following algorithm which implies arithmetic progression $3,5,7,9...$.Each term of rank $k\ge2$ is a difference between the product of the last number on the board and the term of arithmetic progression of rank $k$ and the last but one term on the bord with the sum of the terms of the arithemtic progression with ranks less than $k$.If $x_0=0 $ and $x_1=1$ find $x_n$ according to n.
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 $n$.
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.
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.
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 $13$.
Consider the sequence $1, \frac12, \frac13, \frac14 ,...$
Does there exist an arithmetic progression composed of terms of this sequence
(a) of length $5$,
(b) of length greater than $5$ (if so, what possible length)?
(G Galperin, Moscow)
The numbers $a^2, b^2, c^2$ form an arithmetic progression. Show that the numbers $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ also form arithmetic progression.
Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$
Consider a countinuous function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that verifies the following conditions:
$ \text{(1)} x f(f(x))=(f(x))^2,\quad\forall x\in\mathbb{R}_{>0} $
$ \text{(2)} \lim_{\stackrel{x\to 0}{x>0}} \frac{f(x)}{x}\in\mathbb{R}\cup\{ \pm\infty \} $
[b]a)[/b] Show that $ f $ is bijective.
[b]b)[/b] Prove that the sequences $ \left( (\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}} ) (x) \right)_{n\ge 1} ,\left( (\underbrace{f^{-1}\circ f^{-1}\circ\cdots \circ f^{-1}}_{\text{n times}} ) (x) \right)_{n\ge 1} $ are both arithmetic progressions, for any fixed $ x\in\mathbb{R}_{>0} . $
[b]c)[/b] Determine the function $ f. $
[i]Nelu Chichirim[/i]
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Proposed by Evan Chen[/i]
Source: 1976 Euclid Part B Problem 2
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Given that $x$, $y$, and $2$ are in geometric progression, and that $x^{-1}$, $y^{-1}$, and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$.
The set $S=\{ \frac{1}{n} \; \vert \; n \in \mathbb{N} \}$ contains arithmetic progressions of various lengths. For instance, $\frac{1}{20}$, $\frac{1}{8}$, $\frac{1}{5}$ is such a progression of length $3$ and common difference $\frac{3}{40}$. Moreover, this is a maximal progression in $S$ since it cannot be extended to the left or the right within $S$ ($\frac{11}{40}$ and $\frac{-1}{40}$ not being members of $S$). Prove that for all $n \in \mathbb{N}$, there exists a maximal arithmetic progression of length $n$ in $S$.
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.
[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 $k+1$.
$m$ is an integer satisfying $m \ge 2024$ , $p$ is the smallest prime factor of $m$ , for an arithmetic sequence $\{a_n\}$ of positive numbers with the common difference $m$ satisfying : for any integer $1 \le i \le \frac{p}{2} $ , there doesn’t exist an integer $x , y \le \max \{a_1 , m\}$ such that $a_i=xy$ Try to proof that there exists a positive real number $c$ such that for any $ 1\le i \le j \le n $ , $gcd(a_i , a_j ) = c \times gcd(i , j)$
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
[i]Proposed by Igor Voronovich, Belarus[/i]
Prove that every infinite arithmetic progression $a$, $a+d$, $a+2d$,... where $a$ and $d$ are positive integers, contains infinte geometric progression $b$, $bq$, $bq^2$,... where $b$ and $q$ are also positive integers
Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence.
Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.
Let $q$ and $r$ be integers with $q>0,$ and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B,$ and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T.$ Show that if the product of the lengths of $A$ and $B$ is less than $q,$ then $S$ is the intersection of $A$ with some arithmetic progression.
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$.