An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle? [asy] size(6cm); defaultpen(fontsize(9pt)); path rectangle(pair X, pair Y){ return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle; } filldraw(rectangle((0,0),(7,5)),gray(0.5)); filldraw(rectangle((1,1),(6,4)),gray(0.75)); filldraw(rectangle((2,2),(5,3)),white); label("$1$",(0.5,2.5)); draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead)); draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead)); label("$1$",(1.5,2.5)); draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead)); draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead)); label("$1$",(4.5,2.5)); draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead)); draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead)); label("$1$",(4.1,1.5)); draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead)); draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead)); label("$1$",(3.7,0.5)); draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead)); draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead)); [/asy] $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) }8$

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties? [list] [*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$; [*] $a_2 - a_1$ is not divisible by $m$. [/list] [i]Holden Mui[/i]

Determine all positive integers $n$, $n\ge2$, such that the following statement is true: If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

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]

Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable

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 $(a_n)_{n\ge 1}$ be a sequence of real numbers such that \[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\] Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression. [i]Lucian Dragomir[/i]

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?

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression

The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? $\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $

Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.

Set $S=\{1,2,\cdots,n\}$. $A$ is an increasing arithmetic sequence (at least two numbers), and all numbers are in $S$. Also, we can't add any number in $S$ to $A$ without changing its tolerance. Find the number of such sequence $A$.

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 $M$?

For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.

Is it possible to partition all positive integers into disjoint sets $A$ and $B$ such that (i) no three numbers of $A$ form an arithmetic progression, (ii) no infinite non-constant arithmetic progression can be formed by numbers of $B$?

Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic. [b][i]Alternative formulation[/i][/b] Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression. [i]Proposed by Romania[/i]

For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) \equal{} 1,$ and for every positive integer $ n,$ $ f(n\plus{}1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m \equal{} n$ for which \[ f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m).\] Prove that there are positive integers $ a$ and $ b$ such that $ f(an\plus{}b) \equal{} n\plus{}2$ for every positive integer $ n.$

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

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 $S$.

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 dierence between consecutive terms is constant).

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 $13$.

(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression. (b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.