Found problems: 5802
Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root.
[i]Proposed by Hesam Rajabzadeh [/i]
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.
A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
There is a simple graph which chromatic number is equal to $k$. We painted all of the edges of graph using two colors. Prove that there exist a monochromatic tree with $k$ vertices
Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that
\[\binom{m}{n-1}>\binom{m-1}{n}.\]
Evaluate
\[\lim_{n\to\infty}\frac{M(n)}{n}.\]
The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.
Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.
Devah draws a row of 1000 equally spaced dots on a sheet of paper. She goes through the dots from left to right, one by one, checking if the midpoint between the current dot and some remaining dot to its left is also a remaining dot. If so, she erases the current dot. How many dots does Devah end up erasing?
Let $\{a_n\}_{n\geq 0}$ be a non-decreasing, unbounded sequence of non-negative integers with $a_0=0$. Let the number of members of the sequence not exceeding $n$ be $b_n$. Prove that \[ (a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1). \]
For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence
\[ (r(1),r(3),\ldots,r(1023))
\]
is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$.
Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
A bookshelf contains $n$ volumes, labelled $1$ to $n$, in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label $k$, takes it out, and inserts it in the $k$-th position. For example, if the bookshelf contains the volumes $1,3,2,4$ in that order, the librarian could take out volume $2$ and place it in the second position. The books will then be in the correct order $1,2,3,4$.
(a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order.
(b) What is the largest number of steps that this process can take?
Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.
[i]Proposed by Mahyar Sefidgaran, Iran[/i]
We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.
Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by
\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]
Prove that the following are equivalent:
[list=1]
[*] $f$ is surjective;
[*] $c=0$, $b<d$ and $0<a\le d$.
[/list]
[i]Tiberiu Trif[/i]
Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors.
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
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]
Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
[i]Proposed by Canada[/i]
Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i]