Problems

On an \(8\times 8\) board there is a lamp in every square. Initially every lamp is turned off. In a move we choose a lamp and a direction (it can be the vertical direction or the horizontal one) and change the state of that lamp and all its neighbours in that direction. After a certain number of moves, there is exactly one lamp turned on. Find all the possible positions of that lamp.

In an \(5\times 5\) board one corner was removed. Is it possible to cover the remaining board with linear trominos (\(1\times 3\) blocks)?

We refer to the numeral system that uses base \(2\) as binary. Convert the binary number \(10011_2\) into decimal, and convert the decimal number \(28\) into binary. Multiply by \(2\) the binary number \(10011_2\).

The numeral system that uses base \(3\) is called ternary. Here the only digits are \(0,\) \(1\) and \(2\). Therefore the number \(3\) is written in ternary as \(10_3\). Write down the numbers \(23_{10}\) and \(156_{10}\) in ternary.

Write down the first fifteen natural numbers in binary.

You start with the number \(1\) on a piece of paper. You may perform two operations:

double up (multiply the number on your paper by \(2\), and erase the old number), increase by \(1\) (add \(1\) to the number on your paper, and erase the old number).

So for example, you may end up with the numbers \(1,2,3,6,\cdots\). Show that it is possible to obtain \(975\) starting from \(1\) in at most \(16\) operations.

Alice and Basilio make \(20\) bills. On each bill they will write a seven-digit number, so each bill initially has \(7\) empty cells for digits.

Basilio repeatedly calls out a digit, either \(1\) or \(2\). After each call, Alice writes this digit into any empty cell of any bill and shows the result to Basilio. When all cells are filled, Basilio takes as many bills with different numbers as possible (if several bills have the same number, he takes only one of them), and Alice keeps the rest.

What is the largest number of bills Basilio can guarantee to get, no matter how Alice plays?

The king decided to reward a group of \(n\) wise men. They will be placed in a row one after the other (so that everyone is looking in the same direction), and each is going to wear a black or a white hat. Everyone will see the hats of everyone in front, but not those behind them. The wise men will take turns (from the last to the first) to name the color (white or black) and the natural number of their choice.
At the end, the number of sages who have named the color of their hat correctly is counted: that is exactly how many days the whole group will be paid a salary raise. The wise men were allowed to agree in advance on how to respond. At the same time, the wise men know that exactly \(k\) of them are insane (they do not know who exactly). Any insane man names the color white or black, regardless of the agreement. What is the maximum number of days with a pay supplement that the wise men can guarantee to a group, regardless of the location of the insane in the queue?

A whole number of litres of water is distributed among three vessels. A legal move consists of choosing two vessels and pouring from one into the other exactly as many litres as the receiving vessel already contains. For example, if your vessels contain \((4,6,2)\) liters, then a legal move may be turning this into \((8,2,2)\). The vessels are large enough so that overflow never occurs.

Prove that after finitely many legal moves, one of the vessels can be made empty.

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:

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 \(42\).