Found problems: 1239
Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$
Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then
$$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$
Definitely all positive integers $n$ such that if $a_0 = 1/n$, then this sequence is constant after a certain term.
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$
Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any
positive integers $i, j$:
$$d(a_i + a_j ) = i + j?$$
Here $d(n)$ is the number of positive divisors of a positive integer
We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$
[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes
\[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\]
[b](b)[/b] Find the rearrangement that minimizes $Q.$
A boy lives in a small island in which there are three roads at every junction. He starts
from his home and walks along the roads. At each junction he would choose to turn
to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,
left,... Prove that he will eventually return to his home.
The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$, where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$. For what other d is this true?
Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $
(a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence.
(b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.
Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system:
$$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$
Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $
[i]Florian Dumitrel[/i]
Define the function $f:\mathbb N_{>1}\to\mathbb N_{>1}$ such that $f(x)$ is the greatest prime factor of $x$. A sequence of positive integers $\{a_n\}$ satisfies $a_1=M>1$ and
$$a_{n+1}=\begin{cases}a_n-f(a_n)&\text{if }a_n\text{ is composite.}\\a_n+k&\text{otherwise.}\end{cases}$$
Show that for any positive integers $M,k$, the sequence $\{a_n\}$ is bounded.
(TAN768092100853)
Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges.
Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?
The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
Consider a sequence whose first term is a given positive integer \( N > 1 \). Consider the prime factorization of \( N \). If \( N \) is a power of 2, the sequence consists of a single term: \( N \). Otherwise, the second term of the sequence is obtained by replacing the largest prime factor \( p \) of \( N \) with \( p + 1 \) in the prime factorization. If the new number is not a power of 2, we repeat the same procedure with it, remembering to factor it again into primes. If it is a power of 2, the numerical sequence ends. And so on.
For example, if the first term of the sequence is \( N = 300 = 2^2 \cdot 3 \cdot 5^2 \), since its largest prime factor is \( p = 5 \), the second term is \( 2^2 \cdot 3 \cdot (5 + 1)^2 = 2^4 \cdot 3^3 \). Repeating the procedure, the largest prime factor of the second term is \( p = 3 \), so the third term is \( 2^4 \cdot (3 + 1)^3 = 2^{10} \). Since we obtained a power of 2, the sequence has 3 terms: \( 2^2 \cdot 3 \cdot 5^2 \), \( 2^4 \cdot 3^3 \), and \( 2^{10} \).
a) How many terms does the sequence have if the first term is \( N = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \)?
b) Show that if a prime factor \( p \) leaves a remainder of 1 when divided by 3, then \( \frac{p+1}{2} \) is an integer that also leaves a remainder of 1 when divided by 3.
c) Present an initial term \( N \) less than 1,000,000 (one million) such that the sequence starting from \( N \) has exactly 11 terms.
Let $(a_n)$ be a decreasing sequence of positive numbers with limit $0$ such that
$$b_n = a_n -2 a_{n+1}+a_{n+2} \geq 0$$
for all $n.$ Prove that
$$\sum_{n=1}^{\infty} n b_n =a_1.$$
Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by
$$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$
Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are.
[i]Proposed by Sutanay Bhattacharya[/i]
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$
$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
Find all the sequences of natural $k_n$ with two properties:
a) $k_n \le n \sqrt {n}$ for all $n$
b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$
For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]
Let $R_1 =1$ and $R_{n+1}= 1+ n\slash R_n$ for $n\geq 1.$ Show that for $n\geq 1,$
$$ \sqrt{n} \leq R_n \leq \sqrt{n} +1.$$