Problems

In a room, there are 85 balloons – red and blue ones. It is known that: 1) at least one of the balls is red, 2) from each arbitrarily chosen pair of balls at least one is blue. How many red balls are there in the room?

When Gulliver came to Lilliput, he found that there all things were exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into one of Gulliver’s matchboxes?

If the Humpbacked Horse does not eat or sleep for seven days, it will lose its magical powers. Suppose he did not eat or sleep for a week. What should he do first of all by the end of the seventh day – eat or sleep, so as not to lose powers?

Among some number of mathematicians, every seventh is a philosopher, and among some number of philosophers every ninth is a mathematician. Who are there more of: philosophers or mathematicians?

Find numbers equal to twice the sum of their digits.

Know-it-all came to visit the twin brothers Screw and Nut, knowing that one of them never speaks the truth, and asked one of them: “Are you Screw?”. “Yes,” he replied. When Know-it-all asked the second brother the same question, he received an equally clear answer and immediately determined who was who.

Who was called Screw?

Uncle Jack, the cat Whiskers, Spot and postman Pat are sitting on a bench. If Spot, sitting to the right of everyone, sits between Uncle Jack and the cat, then the cat will be at the extreme left. In what order do they sit?

Cut a square into three pieces, from which you can construct a triangle with three acute angles and three different sides.

A pedestrian walked along six streets of one city, passing each street exactly twice, but could not get around them, having passed each one only once. Could this be?

When Harvey was asked to come up with a problem for the mathematical Olympiad in Sunny City, he wrote a rebus (see the drawing). Can it be solved? (Different letters must match different numbers).