Found problems: 5802
A domino is a $ 1 \times 2 $ or $ 2 \times 1 $ tile.
Let $n \ge 3 $ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some $k \ge 1 $ such that each row and each column has a value of $k$. Prove that a balanced configuration exists for every $n \ge 3 $, and find the minimum number of dominoes needed in such a configuration.
In a rectangular array of nonnegative reals with $m$ rows and $n$ columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that $m=n$.
Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $ 1$ unit to the right and the second one $ 1$ unit to the left.
Under which initial conditions is it possible to move all coins to one single point?
Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.
Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold:
$\begin{array}{rl}
i. & f(x^2) = f(x)^2 -2xf(x) \\
ii. & f(-x) = f(x-1)\\
iii. & 1<x<y \Longrightarrow f(x) < f(y).
\end{array}$
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$
We call natural number $m$ [b]ziba[/b], iff every natural number $n$ with the condition $1\le n\le m$ can be shown as sum of [some of] positive and [u]distinct[/u] divisors of $m$. Prove that infinitely ziba numbers in the form of $(k\in\mathbb{N})k^2+k+2022$ exist.
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.
In an election, there are $1395$ candidates and some voters. Each voter, arranges all the candidates by the priority order.
We form a directed graph with $1395$ vertices, an arrow is directed from $U$ to $V$ when the candidate $U$ is at a higher level of priority than $V$ in more than half of the votes. (otherwise, there's no edge between $U,V$)
Is it possible to generate all complete directed graphs with $1395$ vertices?
Let $\{a_n\}$ be a sequence defined by $a_1=0$ and $$a_n=\frac{1}{n}+\frac{1}{\lceil \frac{n}{2} \rceil}\sum_{k=1}^{\lceil \frac{n}{2} \rceil}a_k$$ for any positive integer $n$. Find the maximal term of this sequence.
A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$, what is left is just $1-2z$. Determine, with proof, $b_{32}$.
Call a positive integer $N \ge 2$ ``special'' if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?
Given a positive integer $k$ and an integer $a\equiv 3 \pmod{8}$, show that $a^m+a+2$ is divisible by $2^k$ for some positive integer $m$.
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
We have $n$ points in the plane, no three on a line.
We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon.
Suppose that for a fixed $k$ the number of $k$ good points is $c_k$.
Show that the following sum is independent of the structure of points and only depends on $n$ :
\[ \sum_{i=3}^n (-1)^i c_i \]
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.
Let $ k>1$ be an integer. Prove that there exists infinitely many natural numbers such as $ n$ such that: \[ n|1^n\plus{}2^n\plus{}\dots\plus{}k^n\]
Zscoder has an simple undirected graph $G$ with $n\ge 3$ vertices. Navi labels a positive integer to each vertex, and places a token at one of the vertex. This vertex is now marked red. In each turn, Zscoder plays with following rule:
$\bullet$ If the token is currently at vertex $v$ with label $t$, then he can move the token along the edges in $G$ (possibly repeating some edges) exactly $t$ times. After these $t$ moves, he marks the current vertex red where the token is at if it is unmarked, or does nothing otherwise, then finishes the turn.
Zscoder claims that he can mark all vertices in $G$ red after finite number of turns, regardless of Navi's labels and starting vertex. What is the minimum number of edges must $G$ have, in terms of $n$?
[i]Proposed by Yeoh Zi Song[/i]
Let $n$ be natural number. Each of the numbers $\in\{1,2,\ldots ,n\}$ is coloured in black or white. When we choose a number, we flip it's colour and the colour of all the numbers which have at least one common divider with the chosen number. At the beginning all the numbers were coloured white. For which $n$ are all the numbers black after a finite number of changes?
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Let $n$ be a positive integer. Suppose we are given $2^n+1$ distinct sets, each containing finitely many objects. Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category. We define the [i]symmetric difference[/i] of two sets as the set of objects belonging to exactly one of the two sets. Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set.
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )