We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (other)

In 25 boxes there are spheres of different colours. It is known that for any $k$ where $1 \leq k \leq 25$ in any $k$ of the boxes there are spheres of exactly $k + 1$ different colours. Prove that a sphere of one particular colour lies in every single box.

Problem #PRU-111923

Problems Calculus Derivative Derivative and extrema Derivative and multiple roots

15–16 3.0

When water is drained from a pool, the water level $h$ in it varies depending on the time $t$ according to the function $h (t) = a t^{2} + b t + c$ , and at the time $t_{0}$ of when the draining is ending, the equalities $h (t_{0}) = h^{'} (t_{0}) = 0$ are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?

Problem #PRU-115374

Problems Set theory and logic Mathematical logic Mathematical logic (other)

10–12 2.0

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.

Problem #PRU-115379

Problems Set theory and logic Theory of algotithms Theory of algorithms (other)

11–12 2.0

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?

Problem #PRU-115384

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

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.

Problem #PRU-115386

Problems Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other) Set theory and logic Theory of algotithms

11–13 2.0

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?

Problem #PRU-115449

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–16 4.0

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an $8 \times 8$ square grid, so that no more ’corners’ would fit?

Problem #PRU-115454

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

13–15 4.0

An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.

Problem #PRU-115467

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–15 4.0

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.

Problem #PRU-115470

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

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?