Problems

A unit square is divided into $n$ triangles. Prove that one of the triangles can be used to completely cover a square with side length $\frac{1}{n}$.

On the sides $AB$, $BC$ and $AC$ of the triangle $ABC$ points $P$, $M$ and $K$ are chosen so that the segments $AM$, $BK$ and $CP$ intersect at one point and $$\overrightarrow{AM}+\overrightarrow{BK}+\overrightarrow{CP}=0$$ Prove that $P$, $M$ and $K$ are the midpoints of the sides of the triangle $ABC$.

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let $C_1$ be the point of intersection, further from the vertex $C$, of the circles constructed from the medians $A{M}_1$ and $B{M}_2$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $A{A}_1$, $B{B}_1$ and $C{C}_1$ intersect at the same point.

In the acute-angled triangle $ABC$, the heights $A{A}_1$ and $B{B}_1$ are drawn. Prove that $A_{1}C\times BC=B_{1}C\times AC$.

Let $A{A}_1$ and $B{B}_1$ be the heights of the triangle $ABC$. Prove that the triangles $A_1{B}_{1}C$ and $ABC$ are similar.