Found problems: 1782
Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $
Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$
Suppose $F$ is a family of finite subsets of $\mathbb{N}$ and for any 2 sets $A,B \in F$ we have $A \cap B \not= \O$.
(a) Is it true that there is a finite subset $Y$ of $\mathbb{N}$ such that for any $A,B \in F$ we have $A\cap B\cap Y \not= \O$?
(b) Is the above true if we assume that all members of $F$ have the same size?
Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$
Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2\minus{}x\minus{}1\equal{}0$. Let $ a_n\equal{}\frac{\alpha^n\minus{}\beta^n}{\alpha\minus{}\beta}$, $ n\equal{}1, 2, \cdots$.
(1) Prove that for any positive integer $ n$, we have $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}a_n$.
(2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n\minus{}2na^n$ for any positive integer $ n$.
Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality
$a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.
Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon.
[i]Proposed by Hesam Rajabzade[/i]
Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$. Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$.
Find all strictly increasing functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that satisfy
\[f(n+f(n))=2f(n)\quad\text{for all}\ n\in\mathbb{N} \]
Starting with the sequence $F_1 = (1,2,3,4, \ldots)$ of the natural numbers further sequences are generated as follows: $F_{n+1}$ is created from $F_n$ by the following rule: the order of elements remains unchanged, the elements from $F_n$ which are divisible by $n$ are increased by 1 and the other elements from $F_n$ remain unchanged. Example: $F_2 = (2,3,4,5 \ldots)$ and $F_3 = (3,3,5,5, \ldots)$. Determine all natural numbers $n$ such that exactly the first $n-1$ elements of $F_n$ take the value $n.$
(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers?
(b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?
Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]
Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$:
\[\sum^{n}_{k=1} x_k = 27\]
and
\[\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.\]
Let $d > 0$ be an arbitrary real number. Consider the set $S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}$. Prove that $S_{n}(d)$ has a maximum element.
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
Let $m$ and $n$ be positive integers and $S$ be a subset with $(2^m-1)n+1$ elements of the set $\{1,2,3,\ldots, 2^mn\}$. Prove that $S$ contains $m+1$ distinct numbers $a_0,a_1,\ldots, a_m$ such that $a_{k-1} \mid a_{k}$ for all $k=1,2,\ldots, m$.
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
Let $a, b$, and $c$ be integers such that $a+b+c$ divides $a^2 +b^2 +c^2$. Prove that there are infinitely many positive integers $n$ such that $a+b+c$ divides $a^n +b^n +c^n$.
The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.
The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$
Find, as a function of $\, n, \,$ the sum of the digits of
\[ 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), \]
where each factor has twice as many digits as the previous one.
Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].