If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation $\textbf{(A) }r^2+r+1=0\qquad\textbf{(B) }r^2-r+1=0\qquad\textbf{(C) }r^4+r^2-1=0$ $\qquad\textbf{(D) }(r+1)^4+r=0\qquad \textbf{(E) }(r-1)^4+r=0$

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences. If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.

Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence. Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.

For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold: (1) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$ (2) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$ (3) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$ Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that \[ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.\]

Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences. How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression $ (b_n)_{n\ge0}$ with $ b_2\equal{}40 \cdot 2009$ are there?

The second and fourth terms of a geometric sequence are $ 2$ and $ 6$. Which of the following is a possible first term? $ \textbf{(A)}\ \minus{}\!\sqrt3 \qquad \textbf{(B)}\ \minus{}\!\frac{2\sqrt3}{3} \qquad \textbf{(C)}\ \minus{}\!\frac{\sqrt3}{3} \qquad \textbf{(D)}\ \sqrt3 \qquad \textbf{(E)}\ 3$

Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $ \frac{1}{6}$, independent of the outcome of any other toss.) $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{2}{9}\qquad \textbf{(C)}\ \frac{5}{18}\qquad \textbf{(D)}\ \frac{25}{91}\qquad \textbf{(E)}\ \frac{36}{91}$

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Prove that, for any integer $a_{1}>1$, there exist an increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, \cdots$ such that \[a_{1}+a_{2}+\cdots+a_{n}\; \vert \; a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\] for all $n \in \mathbb{N}$.

Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold: $\bullet ~~a_1\ge 2$. $\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$ $\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$ Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$

a) We have a geometric sequence of $3$ terms. If the sum of these terms is $26$ , and their sum of squares is $364$ , find the terms of the sequence. b) Suppose that $a,b,c,u,v,w$ are positive real numbers , and each of $a,b,c$ and $u,v,w$ are geometric sequences. Suppose also that $a+u,b+v,c+w$ are an arithmetic sequence. Prove that $a=b=c$ and $u=v=w$ c) Let $a,b,c,d$ be real numbers (not all zero), and let $f(x,y,z)$ be the polynomial in three variables defined by$$f(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d$$.Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric sequence.

Let (i) $a_1,a_2,a_3$ is an arithmetic progression and $a_1+a_2+a_3=18$ (ii) $b_1,b_2,b_3$ is a geometric progression and $b_1b_2b_3=64$ If $a_1+b_1,a_2+b_2,a_3+b_3$ are all positive integers and it is a ageometric progression, then find the maximum value of $a_3$.

Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.

Let be a natural number $ a\ge 2. $ [b]a)[/b] Show that there is no infinite sequence $ \left( k_n \right)_{n\ge 1} $ of pairwise distinct natural numbers greater than $ 1 $ having the property that the sequence $ \left( a^{1/k_n} \right)_{n\ge 1} $ is a geometric progression. [b]b)[/b] Show that there are finite sequences $ \left( l_i \right)_i, $ of any length, of pairwise distinct natural numbers greater than $ 1 $ with the property that $ \left( a^{1/l_i} \right)_{i} $ is a geometric progression. [i]Bogdan Enescu[/i]

[i]Centered hexagonal numbers[/i] are the numbers of dots used to create hexagonal arrays of dots. The first four centered hexagonal numbers are 1, 7, 19, and 37 as shown below: [asy] size(250);defaultpen(linewidth(0.4)); dot(origin^^shift(-12,0)*origin^^shift(-24,0)*origin^^shift(-36,0)*origin); int i; for(i=0; i<360; i=i+60) { dot(1*dir(i)^^2*dir(i)^^3*dir(i)); dot(shift(1/2, sqrt(3)/2)*1*dir(i)^^shift(1/2, sqrt(3)/2)*2*dir(i)); dot(shift(1, sqrt(3))*1*dir(i)); dot(shift(-12,0)*origin+1*dir(i)^^shift(-12,0)*origin+2*dir(i)); dot(shift(-12,0)*origin+sqrt(3)*dir(i+30)); dot(shift(-24,0)*origin+1*dir(i)); } label("$1$", (-36, -5), S); label("$7$", (-24, -5), S); label("$19$", (-12, -5), S); label("$37$", (0, -5), S); label("Centered Hexagonal Numbers", (-18,-10), S);[/asy] Consider an arithmetic sequence 1, $a$, $b$ and a geometric sequence 1,$c$,$d$, where $a$,$b$,$c$, and $d$ are all positive integers and $a+b=c+d$. Prove that each centered hexagonal number is a possible value of $a$, and prove that each possible value of $a$ is a centered hexagonal number.

The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term? $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 225\qquad \textbf{(E)}\ 315$

In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is $\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$

Let $a,b$ be real numbers ($b\ne 0$) and consider the infinite arithmetic sequence $a, a+b ,a +2b , \ldots.$ Show that this sequence contains an infinite geometric subsequence if and only if $\frac{a}{b}$ is rational.

By adding the same constant to $20,50,100$ a geometric progression results. The common ratio is: $ \textbf{(A)}\ \frac53 \qquad\textbf{(B)}\ \frac43\qquad\textbf{(C)}\ \frac32\qquad\textbf{(D)}\ \frac12\qquad\textbf{(E)}\ \frac13 $

Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.

Suppose we have four real numbers $a,b,c,d$ such that $a$ is nonzero, $a,b,c$ form a geometric sequence, in that order, and $b,c,d$ form an arithmetic sequence, in that order. Compute the smallest possible value of $\frac{d}{a}.$ (A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)

Consider $ 2011 $ positive real numbers $ a_1,a_2,\ldots ,a_{2011} . $ If they are in geometric progression, show that there exists a real number $ \lambda $ such that any $ i\in\{ 1,2,\ldots , 1005 \} $ implies $ \lambda =a_i\cdot a_{2012-i} . $ Disprove the converse. [i]Teodor Radu[/i]

A geometric sequence $ (a_n)$ has $ a_1\equal{}\sin{x}, a_2\equal{}\cos{x},$ and $ a_3\equal{}\tan{x}$ for some real number $ x$. For what value of $ n$ does $ a_n\equal{}1\plus{}\cos{x}$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$