Found problems: 1782
Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$
Let $n$ be a positive integer. Starting with the sequence $1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}$, form a new sequence of $n -1$ entries $\frac{3}{4}, \frac{5}{12},\cdots ,\frac{2n -1}{2n(n -1)}$, by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n -2$ entries and continue until the final sequence consists of a single number $x_n$. Show that $x_n < \frac{2}{n}$.
Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$
Let be some rational numbers with the property that their sum, as well as the product of any two of them is integer. Prove that all these are integers.
Let the sum of the first $ n$ primes be denoted by $ S_n$. Prove that for any positive integer $ n$, there exists a perfect square between $ S_n$ and $ S_{n\plus{}1}$.
Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.
For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.
Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).
Sorry for my bad English.
Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative
integers, defined by the following conditions :
$ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$.
$ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$.
$ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$.
Show that if $a_{1}=5$, the $16 \mid a_{81}$.
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
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.
There are $n$ boxes ${B_1},{B_2},\ldots,{B_n}$ from left to right, and there are $n$ balls in these boxes. If there is at least $1$ ball in ${B_1}$, we can move one to ${B_2}$. If there is at least $1$ ball in ${B_n}$, we can move one to ${B_{n - 1}}$. If there are at least $2$ balls in ${B_k}$, $2 \leq k \leq n - 1$ we can move one to ${B_{k - 1}}$, and one to ${B_{k + 1}}$. Prove that, for any arrangement of the $n$ balls, we can achieve that each box has one ball in it.
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$.
Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by
\[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \]
is a decreasing function.
[i]Dan Marinescu et al.[/i]
Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds:
\[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]
Two teams, $ A$ and $ B$, fight for a territory limited by a circumference.
$ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved.
Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas.
Prove that, for every $ n$, team $ B$ has a winning strategy.
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number?
[i]Nanang Susyanto, Jogjakarta[/i]
For each $ x$ in $ [0,1]$, define
\[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\
2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases}
\]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$
Let
$f(x)=\frac{ax+b}{cx+d}$
$F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$)
Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$,
show that $F_n(x)=x$ (for any valid x).
Let $f: \mathbb{N} \to \mathbb{N}$ be a function that satisfies:
\[
f(1) = 2008,
\]
\[
f(4n^2) = 4f(n^2),
\]
\[
f(4n^2 + 2) = 4f(n^2) + 3,
\]
\[
f(4n(n+1)) = 4f(n(n+1)) + 1,
\]
\[
f(4n(n+1) + 3) = 4f(n(n+1)) + 4.
\]
Determine whether there exists a natural number $m$ such that:
\[
1^2 + 2^2 + \dots + m^2 + f(1^2) + \dots + f(m^2) = 2008m + 251.
\]
Determine all positive integers $n$ for which there exists an in finite subset $A$ of the set $\mathbb{N}$ of positive integers such that for all pairwise distinct $a_1,\ldots , a_n \in A$ the numbers $a_1+\ldots +a_n$ and $a_1a_2\ldots a_n$ are coprime.