Let $AB$ and $CD$ be two parallel chordes on hyperbola $y=1/x$. Lines $AC$ and $BD$ intersect axis $Oy$ at points $A_1$ and $D_1$ respectively, and axis $Ox$ - at points $C_1$ and $B_1$ respectively. Prove that the area of $\triangle A_1OC_1$ equals the area of $\triangle D_1OB_1$

For positive real nubers $a,b,c$ find the maximum real number $x$, such that there exist positive numbers $p,q,r$, such that $p+q+r=1$ and $x$ does not exceed numbers $a\frac{p}{q}, b\frac{q}{r}, c\frac{r}{p}$

Find all pairs $(f,h)$ of functions $f,h:\mathbb{R}\rightarrow\mathbb{R}$ such as for all real $x$ and $y$ the equation holds. $$f(x^2+yh(x))=xh(x)+f(xy)$$

Inside the circle $w$ of radius $1$ there are $n$ line segments with total length $2\sqrt{n}$. Prove that there exists a circle such that its center coincides with a center of $w$ and it intersects at least two of line segments.

Find all pairs $(n;p)$ of natural numbers $n$ and prime numbers $p$, satisfying the equation $$p(p-1)=2(n^3+1)$$

Let point $I$ be an incenter of $\triangle ABC$. Ray $AI$ intersects circumcircle of $\triangle ABC$ at point $D$. Circumcircle of $\triangle CDI$ intersects ray $BI$ at ponts $I$ and $K$. Prove that $BK=CK$.

a) In isosceles trapezoid from six line segments ( four sides and two diagonals) three are colored red and three are colored green. Prove that from one of the triples of line segmets of the same colour it is possible to make a triangle. b) Will the previous statement remain for arbitary trapezoid?

$2n$ girls and $2n$ boys danced on the school ball. It's known, that for any pair of girls the amount of boys danced with only one of them equals $n$. Prove that the previous statement is also true for boys.