Found problems: 5923
Prove that in every sequence of $79$ consecutive positive integers written in the decimal system, there is a positive integer whose sum of digits is divisible by $13$.
Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$.
[i]Proposed by Taiki Aiba[/i]
The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.
If the sequence $\{a_n\}$ is defined by \begin{align*}a_1 &= 2,\\ a_{n+1} &= a_n + 2n\qquad (n\geq 1),\end{align*} then $a_{100}$ equals
$\textbf{(A) }9900\qquad
\textbf{(B) }9902\qquad
\textbf{(C) }9904\qquad
\textbf{(D) }10100\qquad
\textbf{(E) }10102$
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 $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
The sequence $a_i$ is defined as $a_1 = 1$ and
\[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\]
for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.
The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.
Let $\{a_n\}$ be a monotonically decreasing sequence of positive real numbers with limit $0$. Let $\{b_n\}$ be a rearrangement of the sequence such that for every non-negative integer $m$, the terms $b_{3m+1}$, $b_{3m+2}$, $b_{3m+3}$ are a rearrangement of the terms $a_{3m+1}$, $a_{3m+2}$, $a_{3m+3}$. Prove or give a counterexample to the following statement: the series $\sum_{n=1}^\infty(-1)^nb_n$ is convergent.
Let $(a_n)_{n\geq 1}$ be a (not necessarily strictly) increasing sequence of positive integers, such that $a_n \leq 1000n^{0.999}$ for every positive integer $n$. Prove that there exist infinitely many positive integers $n$ for which $a_n$ divides $n$.
Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$
The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.
Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials
\(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\).
[list='a']
[*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\).
[*]Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\).
[/list]
The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$.
[b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$.
[b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.
Let $a_n$ be a sequence recursively de fined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$. For what value of the base $b$ we get the sum $1$?
a palindromic number is a number which is unchanged when order of its digits is reversed.
prove that the arithmetic progression 18, 37,.. contains infinitely many palindromic numbers.
We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that
\[
g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}.
\]
a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$
b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$
Eddie assigns each of Jason, Jerry, and Jonathan a different positive integer. The three are each perfectly logical and currently know that their numbers are distinct but don't know each other's numbers. Additionally, if one of them knows the answer to the question they will say so immediately. They have the following conversation listed below in chronological order:
[list]
[*] Eddie: Does anyone know who has the smallest number?
[*] Jason, Jerry, Jonathan (at the same time): I'm not sure.
[*] Jonathan: Now I know who has the smallest number.
[*] Eddie: Does anyone know who has the largest number?
[*] Jason, Jonathan, Jerry (at the same time): I'm not sure.
[*] Jerry: Now I know who has the largest number.
[*] Jason: Wow, our numbers are in an geometric sequence!
[/list]
Find the sum of their numbers.
Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a [i]factor link [/i]of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions:
(1) $x_0=1, x_l=x$;
(2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ .
Meanwhile, we define $l$ as the length of the [i]factor link [/i] $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest [i]factor link[/i] of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
$\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$
Fix positive integers $n,j,k$.How many integer sequences are there of the form $1\le a_1<a_2<\ldots<a_k\le n$,where $a_{i+1}-a_i\ge j$ for all $1\le i\le k-1$.
Find all functions $f:(0,\infty)\to(0,\infty)$ such that
$$f(f(f(x)))+4f(f(x))+f(x)=6x.$$
(a) Does there exist an infinite sequence of real numbers such that the sum of every ten successive numbers is positive, while for every $n$ the sum of the first $10n + 1$ successive numbers is negative?
(b) Does there exist an infinite sequence of integers with the same properties?
(AK Tolpygo)
Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$. Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that
[list]
[*] $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$, and
[*] $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer.
[/list]
15000 years ago Tilif ministry in Persia decided to define a code for $n\geq2$ cities. Each code is a sequence of $0,1$ such that no code start with another code. We know that from $2^{m}$ calls from foreign countries to Persia $2^{m-a_{i}}$ of them where from the $i$-th city (So $\sum_{i=1}^{n}\frac1{2^{a_{i}}}=1$). Let $l_{i}$ be length of code assigned to $i$-th city. Prove that $\sum_{i=1}^{n}\frac{l_{i}}{2^{i}}$ is minimum iff $\forall i,\ l_{i}=a_{i}$