Found problems: 1782
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:
[b]I.[/b] $f(1) = 2$
[b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$
Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]
All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)[/i]
Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$
\[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\
&w(a,b)&-2&-1&0&1&2\\ \hline
&-2&-1&-2&2&-2&-1\\
&-1&-2&4&-4&4&-2\\
a&0&2&-4&12&-4&2\\
&1&-2&4&-4&4&-2\\
&2&-1&-2&2&-2&-1\\ \hline\end{array}\]
For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)
Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that:
$\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$).
[i]Edited by orl.[/i]
Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.
Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a [i]ghost[/i] if it doesn’t appear in $S$.
What is the smallest ghost that is not a multiple of $p$?
[i]Proposed by Guerrero[/i]
For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that
\[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \]
where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
The natural number $n$ can be replaced by $ab$ if $a + b = n$, where $a$ and $b$ are natural numbers. Can the number $2001$ be obtained from $22$ after a sequence of such replacements?
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.
[i]AMM Magazine[/i]
Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a
sequence of operations of this type?
Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
Let $\alpha$ be an irrational number with $0<\alpha < 1$, and draw a circle in the plane whose circumference has length $1$. Given any integer $n\ge 3$, define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b\le n$.
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).
Find all sets $A$ and $B$ that satisfy the following conditions:
a) $A \cup B= \mathbb{Z}$;
b) if $x \in A$ then $x-1 \in B$;
c) if $x,y \in B$ then $x+y \in A$.
[i]Laurentiu Panaitopol[/i]
Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.
Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied:
1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive).
2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd.
3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point.
Determine the maximum possible value that $m$ can take.
Every non-negative integer is coloured white or red, so that:
• there are at least a white number and a red number;
• the sum of a white number and a red number is white;
• the product of a white number and a red number is red.
Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.
Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that:
\[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
[asy]
unitsize(8);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
draw((8,1)--(11,1)--(11,2)--(8,2)--cycle);
draw((19,0)--(20,0)--(20,5)--(19,5)--cycle);
draw((18,1)--(21,1)--(21,4)--(18,4)--cycle);
draw((17,2)--(22,2)--(22,3)--(17,3)--cycle);
draw((32,0)--(33,0)--(33,7)--(32,7)--cycle);
draw((29,3)--(36,3)--(36,4)--(29,4)--cycle);
draw((31,1)--(34,1)--(34,6)--(31,6)--cycle);
draw((30,2)--(35,2)--(35,5)--(30,5)--cycle);
label("Figure",(0.5,-1),S);
label("$0$",(0.5,-2.5),S);
label("Figure",(9.5,-1),S);
label("$1$",(9.5,-2.5),S);
label("Figure",(19.5,-1),S);
label("$2$",(19.5,-2.5),S);
label("Figure",(32.5,-1),S);
label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$.
[i]Hery Susanto, Malang[/i]