Found problems: 5923
Prove that the arithmetic sequence $5, 11, 17, 23, 29, \ldots$ contains infinitely many primes.
Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$.
Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation
\[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \]
if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.
A sequence of integers $a_1, a_2, \ldots, a_n$ is said to be [i]sub-Fibonacci[/i] if $a_1=a_2=1$ and $a_i \le a_{i-1}+a_{i-2}$ for all $3 \le i \le n.$ How many sub-Fibonacci sequences are there with $10$ terms such that the last two terms are both $20$?
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Let be a nonnegative integer $ n. $ Prove that there exists an increasing and finite sequence of positive real numbers, $
\left( a_k \right)_{0\le k\le n} , $ that satisfy the equality
$$ a_0/0! +a_1/1! +a_2/2! +\cdots +a_n/n! =1/n! , $$
and the inequality
$$ a_0+a_1+a_2+\cdots +a_n<\frac{3}{2^n} . $$
[i]Dorin Andrica[/i]
Show that the sequence
\[\binom{2002}{2002},\binom{2003}{2002},\binom{2004}{2002},\ldots \]
considred modulo $2002$, is periodic.
Let $u_1=1,u_2=1$ and for all $k \geq 1$'s
$$u_{k+2}=u_{k+1}+u_{k}$$
Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$
The sequence $x_{n}$ is de fined by:
$x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$
Prove that all members of the sequence are perfect squares.
Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$
\[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\]
where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.
The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$
Suppose there is an infi nite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work:
$\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on.
$\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off.
Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?
It's given system of equations
$a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$
..........
$a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$
such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it
Let $(a_0,a_1,a_2,...)$ and $(b_0,b_1,b_2,...)$ be such sequences of non-negative real numbers, that for every integer $i\geqslant 1$ holds $a_i^2\leqslant a_{i-1}a_{i+1}$ and $b_i^2\leqslant b_{i-1}b_{i+1}$.
Define sequence $c_0,c_1,c_2,...$ as
$$c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.$$
Prove that for every integer $k\geqslant 1$ holds $c_{k}^2\leqslant c_{k-1}c_{k+1}$.
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$.
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Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.
For a word $w$ consisting of $n$ lowercase letters, an [i]edit[/i] is specified by a pair $(i,c)$ where $i\in \{1,\ldots, n\}$ and $c$ is a lowercase letter, and transforms $w$ by replacing its $i^{\text{th}}$ letter with $c$. It is possible that $c$ is the same as the letter it replaced.
How many sequences of six edits transform $\verb|frog|$ into $\verb|goat|$? Note that on the word $\verb|abcd|$, the edits $(1,\verb|a|)$ and $(2,\verb|b|)$ are considered distinct, even though both result in the word $\verb|abcd|$.
Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$.
Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$?
If yes, find the smallest $k$. If not, prove this.
Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.)
a) Prove that $(b_n)_{n \ge 0}$ is unbounded.
b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.
Let $k$ be a positive integer. Prove that there exists an infinite monotone increasing sequence of integers $\{a_{n}\}_{n \ge 1}$ such that \[a_{n}\; \text{divides}\; a_{n+1}^{2}+k \;\; \text{and}\;\; a_{n+1}\; \text{divides}\; a_{n}^{2}+k\] for all $n \in \mathbb{N}$.
Given wo non-negative integers $a$ and $b$, one of them is odd and the other one even. By the following rule we define two sequences $(a_n),(b_n)$:
\[ a_0 = a, \quad a_1 = b, \quad a_{n+1} = 2a_n - a_{n-1} + 2 \quad (n = 1,2,3, \ldots)\]
\[ b_0 = b, \quad b_1 = a, \quad b_{n+1} = 2a_n - b_{n-1} + 2 \quad (n = 1,2,3, \ldots)\]
Prove that none of these two sequences contain a negative element if and only if we have $|\sqrt{a} - \sqrt{b}| \leq 1$.
Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.
The sequence $(a_n)$ is given by $a_1=2$, $a_2=500$, $a_3=2000$ and
$$\frac{a_{n+2}+a_{n+1}}{a_{n+1}+a_{n-1}}=\frac{a_{n+1}}{a_{n-1}}\qquad\text{for }n\ge2$$Prove that all terms of this sequence are positive integers and that $a_{2000}$ is divisible by $2^{2000}$.
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and
${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$
Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.