Found problems: 93
We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.
Show that any arithmetic progression where the first term and the common difference are non-zero natural numbers contains an infinite number of composite terms.
*A number is composite if it is not prime.
Let $a_1, a_2, a_3,\ldots$ and $b_1, b_2, b_3,\ldots$ be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio $a_k/b_k$ is an integer for all k. Is it true that this ratio does not depend on $k{}$?
[i]Boris Frenkin[/i]
Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?
The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.
Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.
Does there exist a number $N$ so that there are $N - 1$ infinite arithmetic progressions with differences $2 , 3 , 4 ,..., N$ , and every natural number belongs to at least one of these progressions?
Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.
Let be a bipartition of the set formed by the first $ 13 $ nonnegative numbers. Prove that at least one of these two subsets that form this partition contains an arithmetic progression.
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 $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$?
[i]Viktor Kleptsyn[/i]
Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.
Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers.
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.
Find all pairs $(a,b)$ of real numbers such that $\lfloor an + b \rfloor$ is a perfect square, for all positive integer $n$.
A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.
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.
An arithmetic progression is formed by $9$ positive integers such that the product of these $9$ terms is a multiple of $3$. Prove that said product is also multiple of $81$.
Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.
Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$
$\textbf{a)}$ Prove that the given sequence is an arithmetic progression.
$\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.
Let $a_1, a_2,...$ a sequence of real numbers.
For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$.
It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$.
Prove that the sequence $a_1, a_2,..$ is arithmetic
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]