A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer. [i]($3$ points)[/i]

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$

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$ Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$

Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.

Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that \[a_1^m+a_2^m+a_3^m+\cdots=m\] for every positive integer $m?$

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.

Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.

Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer. [i]Radu Gologan[/i]

Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

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.

Let $\Gamma_i, i = 0, 1, 2, \dots$ , be a circle of radius $r_i$ inscribed in an angle of measure $2\alpha$ such that each $\Gamma_i$ is externally tangent to $\Gamma_{i+1}$ and $r_{i+1} < r_i$. Show that the sum of the areas of the circles $\Gamma_i$ is equal to the area of a circle of radius $r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).$

If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence $ \textbf{(A)}\ \text{which is a G.P} \qquad$ $ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$ $ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$ $ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$ $ \textbf{(E)}\ \text{none of these}$

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 $n \in \mathbb{N}$.

Given a geometric sequence with the first term $ \neq 0$ and $ r \neq 0$ and an arithmetic sequence with the first term $ \equal{}0$. A third sequence $ 1,1,2\ldots$ is formed by adding corresponding terms of the two given sequences. The sum of the first ten terms of the third sequence is: $ \textbf{(A)}\ 978 \qquad \textbf{(B)}\ 557 \qquad \textbf{(C)}\ 467 \qquad \textbf{(D)}\ 1068 \\ \textbf{(E)}\ \text{not possible to determine from the information given}$

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

$(a_i),(b_i)$ are nonconstant arithmetic and geometric progressions. $a_1=b_1,a_2/b_2=2,a_4/b_4=8$ Find $a_3/b_3$.

A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1$, the terms $a_{2n-1}$, $a_{2n}$, $a_{2n+1}$ are in geometric progression, and the terms $a_{2n}$, $a_{2n+1}$, and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000. Find $n+a_n$.

A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 81$

a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$. b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.

Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles. I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.

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 $a_3$.

Let $ABC$ be an isosceles triangle with $AB=4$, $BC=CA=6$. On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$. On the segment $CB$ consecutively lie points $Y_{1},Y_{2},Y_{3},\ldots$ such that the lengths of the segments $CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. On the segment $AC$ consecutively lie points $Z_{1},Z_{2},Z_{3},\ldots$ such that the lengths of the segments $AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. Find all triplets of positive integers $(a,b,c)$ such that the segments $AY_{a}$, $BZ_{b}$ and $CX_{c}$ are concurrent.

Positive integers $a$, $b$, $c$ are given. It is known that $\frac{c}{b}=\frac{b}{a}$, and the number $b^2-a-c+1$ is a prime. Prove that $a$ and $c$ are double of a squares of positive integers.