Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

Let $n$ be an integer $> 1$. In a circular arrangement of $n$ lamps $L_0, \cdots, L_{n-1}$, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \cdots$. If $L_{j-1}$ ($j$ is taken mod n) is ON, then $Step_j$ changes the status of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If $L_{j-1}$ is OFF, then $Step_j$ does not change anything at all. Show that: [i](a)[/i] There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again. [i](b)[/i] If $n$ has the form $2^k$, then all lamps are ON after $n^2 - 1$ steps. [i](c) [/i]If $n$ has the form $2^k +1$, then all lamps are ON after $n^2 -n+1$ steps.

Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$, $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$. Prove that \[ \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). \]

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.

1. Prove the following inequality for positive reals $a_1,a_2...,a_n$ and $b_1,b_2...,b_n$: $(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})$

Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.

The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$, respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that\[ \frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.\]

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

Let $ n\ge 2$ be an integer. What is the minimal and maximal possible rank of an $ n\times n$ matrix whose $ n^{2}$ entries are precisely the numbers $ 1, 2, \ldots, n^{2}$?

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$