Problems
100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.
A block of cheese comes in packaging with parallel lines of different colours printed on it. If you cut along the red lines then you will get 5 slices of cheese, if you cut along the yellow lines then there will be 7 slices, and along the green lines you will get 11 slices. How many slices will you get if you cut along the lines of all three colours?
In Neverland, only elves and gnomes live. Gnomes lie about their gold, but in any other instances they tell the truth. Elves lie when talking about gnomes, but in other instances they tell the truth. One day two neverlandians said:
\(A\): All my gold I stole from the Dragon.
\(B\): You’re lying.
Determine whether each of them is an elf or a gnome.
Hannah has a calculator that allows you to multiply a number by 3, add 3 to the number or (4 if the number is divisible by 3 to make a whole number) divide by 3. How can the number 11 be made on this calculator from the number 1?
A game of ’Battleships’ has a fleet consisting of one \(1\times 4\) square, two \(1\times 3\) squares, three \(1\times 2\) squares, and four \(1\times 1\) squares. It is easy to
distribute the fleet of ships on a \(10\times
10\) board, see the example below. What is the smallest square
board on which this fleet can be placed? Note that by the rules of the
game, no two ships can be placed on horizontally, vertically, or
diagonally adjacent squares.
In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?
The product of two natural numbers, each of which is not divisible by 10, is equal to 1000. Find the sum of these two numbers.
10 children, including Billy, attended Billy’s birthday party. It turns out that any two children picked from those at the party share a grandfather. Prove that 7 of the children share a grandfather.
An old analogue clock speeds up by 9 minutes after 24 hours. If you went to sleep at 22:00 and set the correct time on the clock, then for what time should the alarm be set if you want it to go off at exactly 6:00? Explain your answer.
A class has 25 pupils. It is known that for any two girls in the class, the number of male friends they have in the class is different. What is the maximum number of girls that it is possible for there to be in the class?