We solve problems

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

One day all the truth tellers on the planet decided to carry a clearly visible mark of truth in order to be distinguished from liars. Two truth tellers and two liars met and looked at each other. Which of them could say the phrase:

“All of us are truth tellers.”
“Only one of you is a truth teller.”
“Exactly two of you are truth tellers.”

Problem #PRU-111328

Problems Fun Problems Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.

Problem #PRU-111345

Problems Calculus Integral Applications of integration Area by integration Algebra Polynomials Quadratic polynomials Quadratic equations. Vieta's rule.

16–16 3.0

The numbers $p$ and $q$ are such that the parabolas $y = - 2 x^{2}$ and $y = x^{2} + p x + q$ intersect at two points, bounding a certain figure.

Find the equation of the vertical line dividing the area of this figure in half.

Problem #PRU-111415

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

13–15 3.0

The volume of the regular quadrangular pyramid $S A B C D$ is equal to $V$ . The height $S P$ of the pyramid is the edge of the regular tetrahedron $S P Q R$ , the plane of the face $P Q R$ which is perpendicular to the edge $S C$ . Find the volume of the common part of these pyramids.

Problem #PRU-111417

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

13–15 3.0

The height $S O$ of a regular quadrilateral pyramid $S A B C D$ forms an angle $α$ with a side edge and the volume of this pyramid is equal to $V$ . The vertex of the second regular quadrangular pyramid is at the point $S$ , the centre of the base is at the point $C$ , and one of the vertices of the base lies on the line $S O$ . Find the volume of the common part of these pyramids.

Problem #PRU-111682

Problems Fun Problems Word Problems Tables and tournaments Chessboards and chess pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Proof by contradiction

12–14 3.0

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.

Problem #PRU-111770

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

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-111805

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

13–15 3.0

The sequence $(a_{n})$ is given by the conditions $a_{1} = 1000000$ , $a_{n + 1} = n ⌊ a_{n} / n ⌋ + n$ . Prove that an infinite subsequence can be found within it, which is an arithmetic progression.

Problem #PRU-111829

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

13–15 3.0

In the infinite sequence $(x_{n})$ , the first term $x_{1}$ is a rational number greater than 1, and $x_{n + 1} = x_{n} + \frac{1}{⌊ x_{n} ⌋}$ for all positive integers $n$ .

Prove that there is an integer in this sequence.

Note that in this problem, square brackets represent integers and curly brackets represent non-integer values or 0.

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?