Found problems: 492
The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$.
[i]Proposed by Lewis Chen[/i]
Is it true that for any positive integer $n>1$, there exists an infinite arithmetic progression $M_n$ of positive integers, such that for any $m \in M_n$, the number $n^m-1$ is not a perfect power (a positive integer is a perfect power if it is of the form $a^b$ for positive integers $a, b>1$)?
For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.
Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties:
(a). $x_1 < x_2 < \cdots < x_{n-1}$ ;
(b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$;
(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
[i]Vasile Pop[/i]
Let $ v$, $ w$, $ x$, $ y$, and $ z$ be the degree measures of the five angles of a pentagon. Suppose $ v < w < x < y < z$ and $ v$, $ w$, $ x$, $ y$, and $ z$ form an arithmetic sequence. Find the value of $ x$.
$ \textbf{(A)}\ 72 \qquad \textbf{(B)}\ 84 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 120$
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 $M < A +\sqrt{B}$.
The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is:
$\textbf{(A) }27\qquad
\textbf{(B) }27\tfrac14\qquad
\textbf{(C) }27\tfrac12\qquad
\textbf{(D) }28\qquad
\textbf{(E) }28\tfrac12$
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.
Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$?
$\text{(A)}$ It has no real roots.
$\text{(B)}$ It has two equal real roots.
$\text{(C)}$ It has two different real roots, and their product is positive.
$\text{(D)}$ It has two different real roots, and their product is negative.
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.
[i]L. Kuptsov[/i]
Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.
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.
The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$, then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$, are in $M$. Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$?
Let $n \geq 3$ be an integer. Let $f$ be a function from the set of all integers to itself with the following property: If the integers $a_1,a_2,\ldots,a_n$ form an arithmetic progression, then the numbers
$$f(a_1),f(a_2),\ldots,f(a_n)$$
form an arithmetic progression (possibly constant) in some order. Find all values for $n$ such that the only functions $f$ with this property are the functions of the form $f(x)=cx+d$, where $c$ and $d$ are integers.
Right triangle $ABC$ has a right angle at $A.$ Points $D$ and $E$ respectively lie on $\overline{AC}$ and $\overline{BC}$ so that $\angle BDA \cong \angle CDE.$ If the lengths $DE,$ $DA,$ $DC,$ and $DB,$ in this order, form an arithmetic sequence of distinct positive integers, then the set of all possible areas of $\triangle ABC$ is a subset of the positive integers. Compute the smallest element in this set that is greater than $1000.$
Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties:
(a). $x_1 < x_2 < \cdots < x_{n-1}$ ;
(b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$;
(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.
Evaluate the sum $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 \cdots + 208 + 209 - 210.$
Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for:
$\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$
$\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$
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.
Numbers from $1$ to $100$ are written on the board. Is it possible to cross $10$ numbers in such way, that we couldn't select 10 numbers from rest which would form arithmetic progression?
For $ p\equal{}1,2,\ldots,10$ let $ S_p$ be the sum of the first $ 40$ terms of the arithmetic progression whose first term is $ p$ and whose common difference is $ 2p\minus{}1$; then $ S_1\plus{}S_2\plus{}\cdots\plus{}S_{10}$ is
$ \textbf{(A)}\ 80000
\qquad \textbf{(B)}\ 80200
\qquad \textbf{(C)}\ 80400
\qquad \textbf{(D)}\ 80600
\qquad \textbf{(E)}\ 80800$
What is the largest possible length of an arithmetic progression of positive integers $ a_{1}, a_{2},\cdots , a_{n}$ with difference $ 2$, such that $ {a_{k}}^{2}\plus{}1$ is prime for $ k \equal{} 1, 2, . . . , n$?