Found problems: 492
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
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$
The real numbers $x$, $y$, $z$, satisfy $0\leq x \leq y \leq z \leq 4$. If their squares form an arithmetic progression with common difference $2$, determine the minimum possible value of $|x-y|+|y-z|$.
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.
Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence
\[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]
forms an arithmetic progression. Prove that the terms of the sequence are equal.
[b]a)[/b] Find the real numbers $ x,y $ such that the set $ \{ x,y \}\cup\left\{ 31/20,29/30,27/40,11/120 \right\} $ contains six elements that can represent an arithmetic progression.
[b]b)[/b] Let be four real numbers in arithmetic progression $ b_1<b_2<b_3<b_4. $ Are there sets $ S $ of $ 6 $ elements that represent an arithmetic progression such that $ \left\{ b_1,b_2,b_3,b_4 \right\}\subset S? $
An arithmetic sequence is a sequence of $(a_1, a_2, \dots, a_n) $ such that the difference between any two consecutive terms is the same. That is, $a_ {i + 1} -a_i = d $ for all $i \in \{1,2, \dots, n-1 \} $, where $d$ is the difference of the progression.
A sequence $(a_1, a_2, \dots, a_n) $ is [i]tlaxcalteca [/i] if for all $i \in \{1,2, \dots, n-1 \} $, there exists $m_i $ positive integer such that $a_i = \frac {1} {m_i}$. A taxcalteca arithmetic progression $(a_1, a_2, \dots, a_n )$ is said to be [i]maximal [/i] if $(a_1-d, a_1, a_2, \dots, a_n) $ and $(a_1, a_2, \dots, a_n, a_n + d) $ are not Tlaxcalan arithmetic progressions.
Is there a maximal tlaxcalteca arithmetic progression of $11$ elements?
Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions:
(a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.
(b) $n \le f(n) \le n+2012$ for all $n$.
Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.
Find all infinite sequences $a_1, a_2, \ldots$ of positive integers satisfying the following properties:
(a) $a_1 < a_2 < a_3 < \cdots$,
(b) there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$,
(c) there are infinitely many $k$ such that $a_k = 2k-1$.
A donkey suffers an attack of hiccups and the first hiccup happens at $\text{4:00}$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $\text{700th}$ hiccup occur?
$\textbf{(A) }$ $15$ seconds after $\text{4:58}$
$\textbf{(B) }$ $20$ seconds after $\text{4:58}$
$\textbf{(C)}$ $25$ seconds after $\text{4:58}$
$\textbf{(D) }$ $30$ seconds after $\text{4:58}$
$\textbf{(E) }$ $35$ seconds after $\text{4:58}$
Alison has an analog clock whose hands have the following lengths: $a$ inches (the hour hand), $b$ inches (the minute hand), and $c$ inches (the second hand), with $a < b < c$. The numbers $a$, $b$, and $c$ are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day: $A$ inches (the hour hand), $B$ inches (the minute hand), and $C$ inches (the second hand). The numbers $A$, $B$, and $C$ (in this order) are consecutive terms of a geometric sequence. What is the value of $\frac{B}{A}$?
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 $n+a_n$.
Let $S_n$ be the sum of the first $n$ term of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$?
$\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers.
[i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.
For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist infinitely many primes that are not contained in $S_{a}$.
For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the di erence between consecutive terms is constant).
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 $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.
If $ a$, $ b$, $ c$, and $ d$ are positive real numbers such that $ a$, $ b$, $ c$, $ d$ form an increasing arithmetic sequence and $ a$, $ b$, $ d$ form a geometric sequence, then $ \frac{a}{d}$ is
$ \textbf{(A)}\ \frac{1}{12} \qquad
\textbf{(B)}\ \frac{1}{6} \qquad
\textbf{(C)}\ \frac{1}{4} \qquad
\textbf{(D)}\ \frac{1}{3} \qquad
\textbf{(E)}\ \frac{1}{2}$
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.
Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence.
(A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).
For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the di erence between consecutive terms is constant).
You are given $n\ge 4$ positive real numbers. It turned out that all $\frac{n(n-1)}{2}$ of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal.
[i](Proposed by Anton Trygub)[/i]