Problems

Michael thinks of a number no less than \(1\) and no greater than \(1000\). Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking \(10\) questions?

In the rebus below, replace the letters with numbers such that the same numbers are represented with the same letter. The asterisks can be replaced with any numbers such that the equations hold.

An explanation of the notation used: the unknown numbers in the third and fourth rows are the results of multiplying 1995 by each digit of the number in the second row, respectively. These third and fourth rows are added together to get the total result of the multiplication \(1995 \times ***\), which is the number in the fifth row. This is an example of a “long multiplication table”.

The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than their product. Determine these numbers.

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.

Four numbers (from 1 to 9) have been used to create two numbers with four-digits each. These two numbers are the maximum and minimum numbers, respectively, possible. The sum of these two numbers is equal to 11990. What could the two numbers be?

Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number than is divisible by 1987.

Prove that there is a power of 3 that ends in 001.

Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

Without calculating the answer to \(2^{30}\), prove that it contains at least two identical digits.