We solve problems

Problem #PRU-110166

Problems Set theory and logic Theory of algotithms Balance puzzles Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

A set of weights has the following properties: It contains $5$ weights, which are all different in weight. For any two weights, there are two other weights of the same total weight. What is the smallest number of weights that can be in this set?

Problem #PRU-110185

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra and arithmetic 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-111000

Problems 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-111245

Problems 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-111328

Problems Algebra and arithmetic 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 and arithmetic 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-111682

Problems Algebra and arithmetic 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-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?