Alexandre and Herculano are at Campanha station waiting for the train. To entertain themselves, they decide to calculate the length of a freight train that passes through the station without changing its speed. When the front of the train passes them, Alexandre starts walking in the direction of the train's movement and Herculano starts walking in the opposite direction. The two walk at the same speed and each of them stops at the moment they cross the end of the train. Alexandre walked 45 meters and Herculano 30 . How long is the train?
In the equilateral triangle [ABC],D is the midpoint of [AC],E and the orthogonal projection of D over [CB] and F is the midpoint of [DE] . Prove that [FB] and [AE] are perpendicular.
Integers 1 to 36 are written in each "Neuro-Millions" bulletin. A bet on "Neuro-Millions" consists of choosing 6 of these 36 numbers. Then, 6 numbers between 1 and 36 are drawn, and these constitute the key to "Neuro-Milh˜oes". A bet is awarded if it does not contain any of the key numbers. How many bets, at least, are necessary to guarantee a prize?
