Two friends went simultaneously from A to B. The first went by bicycle, the second – by car at a speed five times faster than the first. Halfway along the route, the car was in an accident, and the rest of the way the motorist walked on foot at a speed half of the speed of the cyclist. Which of them arrived at B first?
Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?
A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
One of the four angles formed when two straight lines intersect is \(41^{\circ}\). What are the other three angles equal to?
Let \(AA_1\) and \(BB_1\) be the heights of the triangle \(ABC\). Prove that the triangles \(A_1B_1C\) and \(ABC\) are similar.
The bisector of the outer corner at the vertex \(C\) of the triangle \(ABC\) intersects the circumscribed circle at the point \(D\). Prove that \(AD = BD\).
The vertex \(A\) of the acute-angled triangle \(ABC\) is connected by a segment with the center \(O\) of the circumscribed circle. The height \(AH\) is drawn from the vertex \(A\). Prove that \(\angle BAH = \angle OAC\).
The vertex \(A\) of the acute-angled triangle \(ABC\) is connected by a segment with the center \(O\) of the circumscribed circle. The height \(AH\) is drawn from the vertex \(A\). Prove that \(\angle BAH = \angle OAC\).
From an arbitrary point \(M\) lying within a given angle with vertex \(A\), the perpendiculars \(MP\) and \(MQ\) are dropped to the sides of the angle. From point \(A\), the perpendicular \(AK\) is dropped to the segment \(PQ\). Prove that \(\angle PAK = \angle MAQ\).