Filter Problems

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The angle at the top of a crane is 20$^{\circ}$. How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?

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 $\frac{1}{n}$.

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?

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

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 $AM_{1}$ and $BM_{2}$. Points $A_{1}$ and $B_{1}$ are defined similarly. Prove that the lines $AA_{1}$, $BB_{1}$ and $CC_{1}$ intersect at the same point.

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.