Found problems: 5802
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$
Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that
\[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]
Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions:
(1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$
(2) For all $n\ge M$, $f(n)\ge 0$
(3) For all $n>m>M$, $n-m|f(n)-f(m)$
Show that $a$ is a perfect $r$-th power.
In every acyclic graph with 2022 vertices we can choose $k$ of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of $k$.
Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to fi nish in the top right square. It can only move horizontally or vertically on the grid.
On which squares can you put the pawn for the snail to be able to finish its journey?
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]
A square is partitioned in $n^2\geq 4$ rectanles using $2(n-1)$ lines,$n-1$ of which,are parallel to the one side of the square,$n-1$ are parallel to the other side.Prove that we can choose $2n$ rectangles of the partition,such that,for each two of them,we can place the one inside the other (possibly with rotation).
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$
Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
Let $n \geq k$ be positive integers and let $a_1, \dots, a_n$ be a non-increasing list of positive real numbers. Prove that there exists $k$ sets $B_1, \dots, B_k$ which partition the set $\{1, 2, \dots, n\}$ such that $$\min_{1 \le j \le k} \left(\sum_{i \in B_j} a_i \right) \geq \min_{1 \le j \le k} \left(\frac{1}{2k+1-2j} \cdot \sum^n_{i=j} a_i\right).$$
[i]Proposed by Navid Safaei[/i]
Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{
\begin{array}{lcr}
a_n+n,\quad a_n\le n, \\
a_n-n,\quad a_n>n,
\end{array}
\right.
\quad n=1,2,\cdots.$
Find the number of positive integers $r$ satisfying $a_r<r\le 3^{2017}$.
Define the sequence $\{a_i\}$ by $a_0=1$, $a_1=4$, and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$. Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$.
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 $a_{2022}\leq 1$.
There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours.
Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies
of $n$.
[i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]
The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$
(1)Find the general formula of $\{a_n\}$;
(2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.
Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero.
Can Amy prevent Bob’s win?
[i]Maxim Didin, Russia[/i]
An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that:
$ (i)$ Each player plays in each round, and every two players meet at most once.
$ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$.
Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament.
[i]Proposed by Carlos di Fiore, Argentina[/i]