This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 18

2020 Moldova Team Selection Test, 1
Tags: arithmetic progression , geometric progression , progression , number theory
All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)

2021 Sharygin Geometry Olympiad, 9.1
Tags: cevian , geometric progression , geometry , geometric sequence
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.

1935 Moscow Mathematical Olympiad, 007
Tags: arithmetic progression , system of equations , geometric progression , algebra
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 $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.

1982 IMO Longlists, 14
Tags: parametric equation , diophantine equation , geometric progression , polynomial , algebra
Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

1995 Portugal MO, 6
Tags: geometric progression , rational , algebra
Prove that a real number $x$ is rational if and only if the sequence $x, x+1, x+2, x+3, ..., x+n, ...$ contains, at least least three terms in geometric progression.

1996 Greece National Olympiad, 1
Tags: recurrence relation , algebra , geometric progression , sequence , geometric sequence
Let $a_n$ be a sequence of positive numbers such that: i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$ ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$. (a) Prove that $(a_n)$ is a geometric progression. (n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$

2017 India PRMO, 5
Tags: geometric sequence , exponential , algebra , geometric progression
Let $u, v,w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.

2009 Tournament Of Towns, 4
Tags: algebra , geometric progression , arithmetic progression
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?

2017 Singapore MO Open, 4
Tags: sequence , algebra , arithmetic progression , geometric progression , inequalities
Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$

III Soros Olympiad 1996 - 97 (Russia), 10.3
Tags: algebra , geometric progression
An infinite sequence of numbers $a, b, c, d,...$ is obtained by term-by-term addition of two geometric progressions. Can this sequence begin with the following numbers:. a) $1,1,3,5$ ? b) $1,2,3,5$ ? c) $1,2,3, 4$ ?

1982 IMO Shortlist, 4
Tags: algebra , polynomial , diophantine equation , parametric equation , geometric progression
Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

2004 Alexandru Myller, 4
Tags: algebra , geometric sequence , number theory , geometric progression
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]

1982 IMO Shortlist, 15
Tags: calculus , inequality , geometric progression
Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.

1949 Moscow Mathematical Olympiad, 162
Tags: algebra , geometric progression , geometric sequence , combinatorics
Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.

2017 India PRMO, 14
Tags: algebra , geometric progression , floor function , integer part , geometric sequence
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.)

2006 Tournament of Towns, 4
Tags: number theory , geometric progression , arithmetic progression , sequence , integer
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)

III Soros Olympiad 1996 - 97 (Russia), 9.5
Tags: algebra , geometric progression
For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

1982 IMO Longlists, 19
Tags: inequality , geometric progression , calculus
Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.