Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

The sequence $ a_1<a_2<...<a_M$ of real numbers is called a weak arithmetic progression of length $ M$ if there exists an arithmetic progression $ x_0,x_1,...,x_M$ such that: $ x_0 \le a_1<x_1 \le a_2<x_2 \le ... \le a_M<x_M.$ $ (a)$ Prove that if $ a_1<a_2<a_3$ then $ (a_1,a_2,a_3)$ is a weak arithmetic progression. $ (b)$ Prove that any subset of $ \{ 0,1,2,...,999 \}$ with at least $ 730$ elements contains a weak arithmetic progression of length $ 10$.

Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?

$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ? $\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$

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 $n$ $\left |a_n-n^2\right | &lt;\frac{n}{2}$. 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 $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.

Let $f : R^+ \to R$ be a strictly increasing function such that $f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2}$ for all $x,y > 0$. Prove that the sequence $a_n = f(n)$ ($n \in N$) does not contain an infinite arithmetic progression.

The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length $1$, then its perimeter is $a + \sqrt{b}4$, where $ a$ and $b$ are positive integers. What is the value of $a + b$?

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 $A$.

Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$ Prove that \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.\]

Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is [i]nice [/i] if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,

How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression? $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 59 \qquad \textbf{(D) } 89 \qquad \textbf{(E) } 178$

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$? $\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$ [asy] size(3.85cm); label("$X$",(2.5,2.1),N); for (int i=0; i<=5; ++i) draw((i,0)--(i,5), linewidth(.5)); for (int j=0; j<=5; ++j) draw((0,j)--(5,j), linewidth(.5)); void draw_num(pair ll_corner, int num) { label(string(num), ll_corner + (0.5, 0.5), p = fontsize(19pt)); } draw_num((0,0), 17); draw_num((4, 0), 81); draw_num((0, 4), 1); draw_num((4,4), 25); void foo(int x, int y, string n) { label(n, (x+0.5,y+0.5), p = fontsize(19pt)); } foo(2, 4, " "); foo(3, 4, " "); foo(0, 3, " "); foo(2, 3, " "); foo(1, 2, " "); foo(3, 2, " "); foo(1, 1, " "); foo(2, 1, " "); foo(3, 1, " "); foo(4, 1, " "); foo(2, 0, " "); foo(3, 0, " "); foo(0, 1, " "); foo(0, 2, " "); foo(1, 0, " "); foo(1, 3, " "); foo(1, 4, " "); foo(3, 3, " "); foo(4, 2, " "); foo(4, 3, " "); [/asy]

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]

There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression. Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.