BY-MO 2018

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair one number is divisible by another.

For every integer $n\geqslant2$ prove the inequality $$\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2},$$ where $k!=1\cdot2\cdot\ldots\cdot k$ .

The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$ . The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$ . The lines $CB$ and $DE$ meet at $F$ . Prove that $AF$ is an altitude of triangle $ABC$ .

Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$ .) Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and $3\times 1$ without overlapping.

The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$ . It is known that angle $BAD=90$ , the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD. Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$ .

For all positive integers $m$ and $n$ prove the inequality $$|n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1.$$

A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$ , $OB$ and $OC$ are equal to $15$ , $12$ and $20$ , respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle. Find the value of the angle $BAC$ .

A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ( $k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$ 1 and $R$ 2 there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$ 1 with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$ 2 with the columns from $S$ there are odd number of $1's$ . Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.