We solve problems

Problems Set theory and logic Algorithm Theory Algorithm Theory

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 \(B\), \(A \cup % \includegraphics{https://problems-static.s3.eu-west-2.amazonaws.com/production/task_images/700/109909-3.png} B\) is also a faction.

Problem #PRU-109937

Problems Methods Invariants and semi-invariants Invariants Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Algebraic methods Processes and operations Proof by exhaustion

14–16 4.0

We are given a table of size \(n \times n\). \(n-1\) of the cells in the table contain the number \(1\). The remainder contain the number \(0\). We are allowed to carry out the following operation on the table:

1. Pick a cell.

2. Subtract 1 from the number in that cell.

3. Add 1 to every other cell in the same row or column as the chosen cell.

Is it possible, using only this operation, to create a table in which all the cells contain the same number?

Problem #PRU-109946

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations Calculus Real numbers Integer and fractional parts. Archimedean property Polynomials Quadratic polynomials Investigating the quadratic polynomial

14–15 3.0

Solve the equation \((x + 1)^3 = x^3\).

Problem #PRU-110047

Problems Algebra 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 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 Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Algebra 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 Algebra 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-110122

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Calculus Functions of one variable. Continuity Monotonicity, boundedness

14–16 4.0

The functions \(f (x) - x\) and \(f (x^2) - x^6\) are defined for all positive \(x\) and increase. Prove that the function

also increases for all positive \(x\).

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 \(\cos a + \cos b + \cos c > \sin a + \sin b + \sin c\).