Problems

Can three points with integer coordinates be the vertices of an equilateral triangle?

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.

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 \(1\).

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?

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 were poured into three vessels. You can only to pour into any vessel the exact amount of water equal to the amount it already contains from any other vessel. Prove that in a few transfusions one can empty one of the vessels. The vessels are large enough: each can hold all the water.

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.