MathDB

2

Part of 2024 Romania National Olympiad

Problems(4)

Romanian National Olympiad 2024 - Grade 9 - Problem 2

Source: Romanian National Olympiad 2024 - Grade 9 - Problem 2

4/6/2024
Let aa and bb be two numbers in the interval (0,1)(0,1) such that aa is rational and {na}≥{nb},\{na\} \ge \{nb\}, for every nonnegative integer n.n. Prove that a=b.a=b.
(Note: {x}\{x\} is the fractional part of x.x.)
algebrafractional part
Romanian National Olympiad 2024 - Grade 10 - Problem 2

Source: Romanian National Olympiad 2024 - Grade 10 - Problem 2

4/6/2024
We consider the inscriptible pentagon ABCDEABCDE in which AB=BC=CDAB=BC=CD and the centroid of the pentagon coincides with the circumcenter. Prove that the pentagon ABCDEABCDE is regular.
The centroid of a pentagon is the point in the plane of the pentagon whose position vector is equal to the average of the position vectors of the vertices.
geometrycircumcirclepentagoninscriptibleCentroid
Romanian National Olympiad 2024 - Grade 11 - Problem 2

Source: Romanian National Olympiad 2024 - Grade 11 - Problem 2

4/4/2024
Let A∈Mn(R)A \in \mathcal{M}_n(\mathbb{R}) be an invertible matrix.
a) Prove that the eigenvalues of AATAA^T are positive real numbers. b) We assume that there are two distinct positive integers, pp and qq, such that (AAT)p=(ATA)q.(AA^T)^p=(A^TA)^q. Prove that AT=A−1.A^T=A^{-1}.
linear algebramatrix
Easy problem about commutativity in division rings

Source: Romanian National Olympiad 2024 - Grade 12 - Problem 2

4/3/2024
Let (K,+,⋅)(\mathbb{K},+, \cdot) be a division ring in which x2y=yx2,x^2y=yx^2, for all x,y∈K.x,y \in \mathbb{K}. Prove that (K,+,⋅)(\mathbb{K},+, \cdot) is commutative.
division ringsabstract algebra