We solve problems

Problem #PRU-110185

Problems Methods Pigeonhole principle Pigeonhole principle (other) Fun Problems Word Problems Tables and tournaments Tournament tables

12–14 3.0

Five teams participated in a football tournament. Each team had to play exactly one match with each of the other teams. Due to financial difficulties, the organisers cancelled some of the games. As a result, it turned out that all teams scored a different number of points and no team scored zero points. What is the smallest number of games that could be played in the tournament, if three points were awarded for a victory, one for a draw and zero for a defeat?

Problem #PRU-110207

Problems Methods Pigeonhole principle Pigeonhole principle (other) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality algebra problems

12–14 3.0

Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.

Problem #PRU-110210

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

13–15 3.0

Prove that for each $x$ such that $\sin x \neq 0$ , there is a positive integer $n$ such that $| \sin n x | \geq \sqrt{3} / 2$ .

Problem #PRU-110212

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

13–15 3.0

For what natural numbers $n$ are there positive rational but not whole numbers $a$ and $b$ , such that both $a + b$ and $a^{n} + b^{n}$ are integers?

Problem #PRU-111000

Problems Discrete Mathematics Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

Problem #PRU-111218

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

13–15 3.0

The base of the pyramid is a square. The height of the pyramid crosses the diagonal of the base. Find the largest volume of such a pyramid if the perimeter of the diagonal section containing the height of the pyramid is 5.

Problem #PRU-111245

Problems Discrete Mathematics Set theory and logic

16–16 2.0

Let’s denote any two digits with the letters $A$ and $X$ . Prove that the six-digit number $X A X A X A$ is divisible by 7 without a remainder.

Problem #PRU-111264

Problems Calculus Functions of one variable. Continuity Continuity and compactness Monotonicity, boundedness

15–16 4.0

A continuous function $f (x)$ is such that for all real $x$ the following inequality holds: $f (x^{2}) - (f (x))^{2} \geq 1 / 4$ . Is it true that the function $f (x)$ necessarily has an extreme point?

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.