Found problems: 93
Is it possible to complete the following square knowning that each row and column make an aritmetic progression?
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.
An infinite arithmetic progression is given. The products of the pairs of its members are considered. Prove that two of these numbers differ by no more than 1.
[i]Proposed by A. Kuznetsov[/i]
Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.
Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.
Let $a_{1},a_{2},\ldots$ be an infinite arithmetic progression. It's known that there exist positive integers $p,q,t$ such that $a_{p}+tp=a_{q}+tq$. If $a_{t}=t$ and the sum of the first $t$ numbers in the sequence is $18$, determine $a_{2008}$.
Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then
$$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$
Alice and Bob play a game on the infinite side of a checkered strip, in which the cells are numbered with consecutive integers from left to right (..., $-2$, $-1$, $0$, $1$, $2$, ...). Alice in her turn puts one cross in any free cell, and Bob in his turn puts zeros in any 2020 free cells. Alice will win if he manages to get such 4 cells marked with crosses, the corresponding cell numbers will form an arithmetic progression. Bob's goal in this game is to prevent Alice from winning. They take turns and Alice moves first. Will Alice be able to win no matter how Bob plays?
Let $a$ be a given positive integer. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n=\frac{1}{1+na},$ for every positive integer $n.$
Prove that for any integer $k \ge 3,$ there exist positive integers $n_1<n_2<\ldots<n_k$ such that the numbers $x_{n_1},x_{n_2},\ldots,x_{n_k}$ are consecutive terms in an arithmetic progression.
Decide if it is possible to generate an infinite sequence of positive integers $a_n$ such that in the sequence there are no three terms that are in arithmetic progression and that for all $n$ $\left |a_n-n^2\right | <\frac{n}{2}$.
Clarification: Three numbers $a$, $b$, $c$ are in arithmetic progression if and only if $2b=a+c$.
Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$
Consider an in finite sequence consisting of distinct positive integers such that each term (except the rst one) is either an arithmetic mean or a geometric mean of two neighboring terms. Does it necessarily imply that starting at some point the sequence becomes either arithmetic progression or a geometric progression?
The lengths of the sides of a certain triangle and the diameter of the inscribed part circles are four consecutive terms of arithmetic progression. Find all such triangles.
Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression.
Given a triangle in which the sides $ a $, $ b $, $ c $ form an arithmetic progression and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle.
The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?
Given a polygon with $n$ sides, we assign the numbers $0,1,...,n-1$ to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for $n = 5$. Notice that the numbers assigned to the sides are still in arithmetic progression.
[img]https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png[/img]
$\bullet$ Make the respective assignment for a $9$-sided polygon, and generalize for odd $n$.
$\bullet$ Prove that this is not possible if $n$ is even.
The equation $x^2 + bx + c = 0$ has two different roots $x_1$ and $x_2$. It is also known that the numbers $b$, $x_1$, $c$, $x_2$ in the indicated order form an arithmetic progression. Find the difference of this progression.
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 $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.
Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?
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]
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.