Problems

Alice the fox and Basilio the cat have grown $20$ counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has $7$ empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?

Cut a $7\times 7$ square into $9$ rectangles, out of which you can construct any rectangle whose sidelengths are less than $7$. Show how to construct the rectangles.

Convert the binary number $10011$ into decimal, and convert the decimal number $28$ into binary. Multiply by $2$ as binary numbers both $10011$ and the result of conversion of $28$ into binary numbers.

The ternary numeral system has only $3$ digits: $0,$ $1$ and $2$. Therefore the number $3$ is written in ternary as $10$. Write down the numbers $23$ and $156$ in ternary and add them as ternary.

Write down the first fifteen numbers in binary system.

Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written as combinations of letters from the Latin alphabet, each letter with a fixed integer value:

I&V&X&L&C&D&M
1&5&10&50&100&500&1000

For example the first $12$ numbers in Roman Numerals are written as: $I,II,III,IV,V,VI,VII,VIII,IX,X,XI,XII$, where the notations $IV$ and $IX$ can be read as "one less than five" and "one less than ten" correspondingly. A number containing two or more decimal digits is built by appending the Roman numeral equivalent for each digit, from highest to lowest, as in the following examples: the current year $2024$ as $MMXXIV$, number $17$ as $XVII$ and number $42$ as $XLII$ or $XXXXII$. Let’s see how to multiply Roman numerals by multiplying $17$ and $42$.

Write down in Roman numerals the numbers $14$ and $61$ and multiply them as Roman numerals.

There are $16$ cities in the kingdom. Prove that it is not possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most four roads coming out of each city.

Given a natural number $n$ you are allowed to perform two operations: "double up", namely get $2n$ from $n$, and "increase by $1$", i.e. to get $n+1$ from $n$. Find the smallest amount of operations one needs to perform to get the number $n$ from $1$.

Detective Nero Wolf investigates a crime. He’s got $80$ people involved in the case, among whom one is a criminal and another is a witness to the crime (but it is not known who either of them are). Each day the detective may invite one or more of these $80$ people, and if there is a witness among those invited, but not the perpetrator, the witness will report who the perpetrator is. Can the detective solve a case in $12$ days?