We solve problems

Problem #PRU-111923

Problems Calculus Derivative Derivative and extrema Derivative and multiple roots

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-111925

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

On the plane coordinate axes with the same but not stated scale and the graph of the function $y = \sin x$ , $x$ $(0; α)$ are given.

How can you construct a tangent to this graph at a given point using a compass and a ruler if: a) $α \in (π / 2; π)$ ; b) $α \in (0; π / 2)$ ?

Problem #PRU-115373

Problems Algebra and arithmetic Arithmetics. Mental maths

10–11 2.0

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?

Problem #PRU-115374

Problems Discrete Mathematics 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 Discrete Mathematics Algorithm Theory 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 Discrete Mathematics 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 Fun Problems Word Problems Tables and tournaments Tables and tournaments (other) Discrete Mathematics Algorithm Theory

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-115397

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

The sequence $a_{1}, a_{2}, \dots$ is such that $a_{1} \in (1, 2)$ and $a_{k + 1} = a_{k} + \frac{k}{a_{k}}$ for any positive integer $k$ . Prove that it cannot contain more than one pair of terms with an integer sum.The sequence $a_{1}, a_{2}, \dots$ is such that $a_{1} \in (1, 2)$ and $a_{k + 1} = a_{k} + \frac{k}{a_{k}}$ for any positive integer $k$ . Prove that it cannot contain more than one pair of terms with an integer sum.

Problem #PRU-115447

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Prove that if the expression

takes a rational value, then the expression

also takes on a rational value.

Problem #PRU-115449

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