Problems

For how many pairs of numbers \(x\) and \(y\) between \(1\) and \(100\) is the expression \(x^2 + y^2\) divisible by \(7\)?

Seven robbers are dividing a bag of coins of various denominations. It turned out that the sum could not be divided equally between them, but if any coin is set aside, the rest could be divided so that every robber would get an equal part. Prove that the bag cannot contain \(100\) coins.

Show that the equation \(x^2 +6x-1 = y^2\) has no solutions in integer \(x\) and \(y\).

Propose a method for measuring the diagonal of a conventional brick, which is easily realied in practice (without the Pythagorean theorem).

The seller with weights. With four weights the seller can weigh any integer number of kilograms, from 1 to 40 inclusive. The total mass of the weights is 40 kg. What are the weights available to the seller?

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).

We are looking for the correct statement. In a notebook one hundred statements are written:

1) There is exactly one false statement in this notebook.

2) There are exactly two false statements in this notebook.

...

100) There are exactly one hundred false statements in this notebook.

Which of these statements is true, if it is known that only one is true?

Solve the rebus \(AC \times CC \times K = 2002\) (different letters correspond to different integers and vice versa).

Rebus. Solve the numerical rebus \(AAAA-BBB + SS-K = 1234\) (different letters correspond to different numbers, but the same letters each time correspond to the same numbers)