Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits $a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$.

Given is the triangle $ABC$, with $| AB | + | AC | = 3 \cdot | BC | $. Let's denote $D, E$ also points that $BCDA$ and $CBEA$ are parallelograms. On the sides $AC$ and $AB$ sides, $F$ and $G$ are selected respectively so that $| AF | = | AG | = | BC |$. Prove that the lines $DF$ and $EG$ intersect at the line segment $BC$

Prove that for all real numbers $x, y$ holds $(x^2 + 1)(y^2 + 1) \ge 2(xy - 1)(x + y)$. For which integers $x, y$ does equality occur?

Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.

Each field of the table $(mn + 1) \times (mn + 1)$ contains a real number from the interval $[0, 1]$. The sum the numbers in each square section of the table with dimensions $n x n$ is equal to $n$. Determine how big it can be sum of all numbers in the table.

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Decide if exists a convex hexagon with all sides longer than $1$ and all nine of its diagonals are less than $2$ in length.

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

Bolek draw a trapezoid $ABCD$ trapezoid ($AB // CD$) on the board, with its midsegment line $EF$ in it. Point intersection of his diagonal $AC, BD$ denote by $P,$ and his rectangular projection on line $AB$ denote by $Q$. Lolek, wanting to tease Bolek, blotted from the board everything except segments $EF$ and $PQ$. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?

In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called balanced if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.

On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.