We solve problems

Problem #PRU-109881

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

13–15 3.0

The polynomial \(P (x)\) of degree \(n\) has \(n\) distinct real roots.

What is the largest number of its coefficients that can be equal to zero?

Problem #PRU-109909

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

10–12 2.0

Members of the State parliament formed factions in such a way that for any two factions \(A\) and \(B\) (not necessarily different)

– also a faction (through

the set of all parliament members not included in \(C\) is denoted). Prove that for any two factions \(A\) and \(B\), \(A \cup % \includegraphics{https://problems-static.s3.amazonaws.com/production/task_images/700/109909-3.png} B\) is also a faction.

Problem #PRU-110047

Problems Algebra and arithmetic Rational and irrational numbers

13–15 3.0

We call a number \(x\) rational if it can be represented as \(x=\frac{p}{q}\) for coprime integers \(p\) and \(q\). Otherwise we call the number irrational.
Non-zero numbers \(a\) and \(b\) satisfy the equality \(a^2b^2 (a^2b^2 + 4) = 2(a^6 + b^6)\). Prove that at least one of them is irrational.

Problem #PRU-110061

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.

Problem #PRU-110072

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?

Problem #PRU-110083

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

Can the cells of a \(5 \times 5\) board be painted in 4 colours so that the cells located at the intersection of any two rows and any two columns are painted in at least three colours?

Problem #PRU-110103

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange the numbers 1, 2, ..., 60 in a circle in such an order that the sum of every two numbers, between which lies one number, is divisible by 2, the sum of every two numbers between which lie two numbers, is divisible by 3, the sum of every two numbers between which lie six numbers, is divisible by 7?

Problem #PRU-110153

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

13–15 3.0

The sum of the positive numbers \(a, b, c\) is \(\pi / 2\). Prove that \(\cos a + \cos b + \cos c > \sin a + \sin b + \sin c\).

Problem #PRU-110166

Problems Discrete Mathematics Algorithm Theory 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) 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?