Problems

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 a nonzero number that is equal to twice the sum of its 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).

In two purses lie two coins, and one purse has twice as many coins as the other. How can this be?

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?

Alex laid out an example of an addition of numbers from cards with numbers on them and then swapped two cards. As you can see, the equality has been violated. Which cards did Alex rearrange?