Found problems: 1782
Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.
In some finite set of positive numbers, each number is expressed
as a linear combination of the rest with rational non-negative coefficients. Prove that the ratio of some two numbers in the set is rational.
Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions:
(a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.
(b) $n \le f(n) \le n+2012$ for all $n$.
Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.
There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that
\[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]
Let's say we have a [i]nice[/i] representation of the positive integer $ n$ if we write it as a sum of powers of 2 in such a way that there are at most two equal powers in the sum (representations differing only in the order of their summands are considered to be the same).
a) Write down the 5 nice representations of 10.
b) Find all positive integers with an even number of nice representations.
Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that
\[f(xf(y)+f(x)) = 2f(x)+xy\]
for every reals $x,y$.
Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game.
For each positive integer $n$, find a winning strategy for one of the players.
Let $S=\{(a,b)|a=1,2,\dots,n,b=1,2,3\}$. A [i]rook tour[/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\dots,p_{3n}$ is sequence such that
(i) $p_i\in S,$
(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\le i<3n,$
(iii) for each $p\in S$ there is a unique $i$ such that $p_i=p.$
How many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$
(The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)
In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.
Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles.
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
Let $a,b,c$ be real numbers such that $ab\not= 0$ and $c>0$. Let $(a_{n})_{n\geq 1}$ be the sequence of real numbers defined by: $a_{1}=a, a_{2}=b$ and
\[a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}\]
for all $n\geq 2$.
Show that all the terms of the sequence are integer numbers if and only if the numbers $a,b$ and $\frac{a^{2}+b^{2}+c}{ab}$ are integers.
An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.
Let $n\ge3$ be a positive integer. Each edge of a complete graph $K_n$ is assigned a real number satisfying the following conditions:
$(i)$ For any three vertices, the numbers assigned to two of the edges among them are equal, and the number on the third edge is strictly greater.
$(ii) $ The weight of a vertex is defined as the sum of the numbers assigned to the edges emanating from that vertex. The weights of all vertices are equal.
Find all possible values of $n$.
There are a buch of 2000 stones. Two players play alternatively, following the next rules:
($a$)On each turn, the player can take 1, 2, 3, 4 or 5 stones [b]of[/b] the bunch.
($b$) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play.
The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.
The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?
[i]Brian Hamrick.[/i]
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$
In a chess tournament there are $n \geq 5$ players, and they have already played $\left[ \frac{n^2}{4} \right] +2$ games (each pair have played each other at most once).
[b](a)[/b] Prove that there are five players $a, b, c, d, e$ for which the pairs $ab, ac, bc, ad, ae, de$ have already played.
[b](b)[/b] Is the statement also valid for the $\left[ \frac{n^2}{4} \right] +1$ games played?
Make the proof by induction over $n.$
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.
[list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$.
[b](b)[/b] Determine all $n$ for which this is possible.[/list]
Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition
\[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\]
Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.