Found problems: 5923
For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]
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] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$.
Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.
The diagram below shows a sequence of equally spaced parallel lines with a triangle whose vertices lie on these lines. The segment $\overline{CD}$ is $6$ units longer than the segment $\overline{AB}$. Find the length of segment $\overline{EF}$.
[img]https://cdn.artofproblemsolving.com/attachments/8/0/abac87d63d366bf4c4e913fdb1022798379a73.png[/img]
Let $n \geq 2$ be a positive integer. Call a sequence $a_1, a_2, \cdots , a_k$ of integers an $n$[i]-chain[/i] if $1 = a_2 < a_ 2 < \cdots < a_k =n$, $a_i$ divides $a_{i+1}$ for all $i$, $1 \leq i \leq k-1$. Let $f(n)$ be the number of $n$[i]-chains[/i] where $n \geq 2$. For example, $f(4) = 2$ corresponds to the $4$-chains $\{1,4\}$ and $\{1,2,4\}$.
Prove that $f(2^m \cdot 3) = 2^{m-1} (m+2)$ for every positive integer $m$.
Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$
The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$,$$n(n+1)
a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$
Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions
$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$
Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.
Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by
\[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\]
and $x_0,y_0$ are given real numbers.
a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit.
b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.
In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$. Prove that $0.99<x_{100}<0.991$.
Fresh translation. This problem may be similar to one of the 9th grade problems.
Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules:
i) Any incantation can appear no more than once;
ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it;
iii)The first person who cannot spell an incantation loses the contest. Answer the following questions:
a) If A says '$STAGEPREIMO$' first, then who will win?
b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.
[b]interesting sequence[/b]
$n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties:
it is periodic and its least period number is $2^n-1$. (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.)
There exist distinct integers $0\le t_1<t_2<...<t_k<n$ such that for every natural number $j$ we have
\[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\]
Prove that for every natural number $s$ that $s<2^n-1$ we have
\[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\]
Time allowed for this question was 1 hours and 15 minutes.
Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$
Show that $r = 2$ is the largest real number $r$ which satisfies the following condition:
If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities
\[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]
for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.
A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?
Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.
A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$,
\[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\]
Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.
Messages are coded using sequences consisting of zeroes and ones only. Only sequences with at most two consecutive ones or zeroes are allowed. (For instance the sequence $011001$ is allowed, but $011101$ is not.) Determine the number of sequences consisting of exactly $12$ numbers.
Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $
[b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $
[b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.