9.5
Problems(3)
How to hang a picture on 2/3 nails (III Soros Olympiad 1996-97 R1 9.5)
Source:
5/29/2024
How to hang a picture? What a strange question? It's simple. We take a piece of rope, attach its ends to the picture frame on the back side, then drive it into the wall. nail and throw a rope over the nail. The picture is hanging. If you pull out the nail, then, of course, it will fall. But Professor No wonder acted differently. At first, he attached the rope to the painting in the same way, only he took it a little longer. Then he hammered two nails into the wall nearby and threw a rope over these nails in a special way. The painting hangs on these nails, but if you pull out any nail, the painting will fall. Moreover, the professor claims that he can hang a painting on three nails so that the painting hangs on all three, but if any nail is pulled out, the painting will fall. You have two tasks: indicate how you can hang the picture in the right way on
a) two nails;
b) three nails.
geometryalgebra
shortest path, square, right isosceles (III Soros Olympiad 1996-97 R2 9.5)
Source:
5/31/2024
An ant sits at vertex of unit square . He needs to get to point , where the entrance to the anthill is located. Points and are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse . Find the length of the shortest path that an ant must overcome in order to get into the anthill.
geometrygeometric inequality
n 7-digit numbers in geom. progression (III Soros Olympiad 1996-97 R3 9.5)
Source:
5/31/2024
For what largest are there seven-digit numbers that are successive members of one geometric progression?
algebrageometric progression