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#### Auxiliary simialr triangles , Extremal principle (other) , Pigeonhole principle (area and volume)

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 $\frac{1}{n}$.

#### The triangle generated by bases of two altitudes and a vertex

Let $AA_1$ and $BB_1$ be the heights of the triangle ABC. Prove that the triangles $A_1B_1C$ and ABC are similar. What is the similarity coefficient?

#### The triangle generated by bases of two altitudes and a vertex

In the acute-angled triangle ABC, the heights $AA_1$ and $BB_1$ are drawn. Prove that $A_1C \times BC$ = $B_1C \times AC$.

#### Ceva's theorem and Menelaus's theorem , Existence of a definite integral , Law of polygon of vectors , Similar triangles (other)

On the sides AB, BC and AC of the triangle ABC points P, M and K are taken so that the segments AM, BK and CP intersect at one point and

Prove that P, M and K are the midpoints of the sides of the triangle ABC.

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