Problems

In an $n\times n$ board the squares are painted black or white in some way. Three of the squares in the corners are white and one is black. Show that there is a $2\times 2$ square with an odd number of white unit squares.

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)?

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.

Two lines $CD$ and $CB$ are tangent to a circle with the center $A$ and radius $R$, see the picture. The angle $\mathrm{\angle }BCD$ equals ${120}^{\circ }$. Find the length of $BD$ in terms of $R$.

Given two circles, one has centre $A$ and radius $r$, another has centre $C$ and radius $R$. Both circles are tangent to a line at the points $B$ and $D$ respectively and the angles $\mathrm{\angle }CED=\mathrm{\angle }AEB={30}^{\circ }$. Find the length of $AC$ in terms of $r$ and $R$.