Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers? (Folklore)

Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.

The six-digit decimal number contains six different non-zero digits and is divisible by $37$. Prove that having transposed its digits you can obtain at least $23$ more numbers divisible by $37$

Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

Let $a, b$ and $c$ be integers such that $a + b + c = 0$. (a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$. (b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$. .

Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.

Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.

How many functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\}$ have the property that $f(f(x))-4x$ is divisible by $17$ for all integers $1 \le x \le 16$?

Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?

For which $m > 1$ is $(m -1)!$ divisible by $m$?

Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

On the squares $a1, a2,... , a8$ of a chessboard there are respectively $2^0, 2^1, ..., 2^7$ grains of oat, on the squares $b8, b7,..., b1$ respectively $2^8, 2^9, ..., 2^{15}$ grains of oat, on the squares $c1, c2,..., c8$ respectively $2^{16}, 2^{17}, ..., 2^{23}$ grains of oat etc. (so there are $2^{63}$ grains of oat on the square $h1$). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by $3$.

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

Let $a,b,c,d$ be integers. Prove that $ 12$ divides $ (a-b) (a-c) (a-d) (b- c) (b-d) (c-d)$.

Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.

a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.

A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.

Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$ where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$. Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi