MathDB

P1

Part of 2024 District Olympiad

Problems(4)

Quadrilateral is parallelogram

Source: Romanian District Olympiad 2024 9.1

3/10/2024
Let ABCDABCD be a convex quadrilateral whose diagonals intersect at O.O. Given that AB→+AD→+AO→=BC→+DC→+OC→,\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AO}=\overrightarrow{BC}+\overrightarrow{DC}+\overrightarrow{OC},prove that ABCDABCD is a parallelogram.
geometryvector
least possible alpha

Source: Romanian District Olympiad, grade 10, p1

3/10/2024
Let a,b∈R, a>1, b>0.a,b\in\mathbb{R},~a>1,~b>0. Find the least possible value for α\alpha such that :(a+b)x≥ax+b, (∀)x≥α.(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.
algebrafunction
Simple 2x2 matrix

Source: Romanian District Olympiad 2024 11.1

3/10/2024
Consider the matrix X∈M2(C)X\in\mathcal{M}_2(\mathbb{C}) which satisfies X2022=X2023.X^{2022}=X^{2023}. Prove that X2=X3.X^2=X^3.
linear algebramatrix
Trivial NT with a ring

Source: Romanian District Olympiad 2024 12.1

3/10/2024
Determine the integers n⩾2n\geqslant 2 for which the equation x2−3^⋅x+5^=0^x^2-\hat{3}\cdot x+\hat{5}=\hat{0} has a unique solution in (Zn,+,⋅).(\mathbb{Z}_n,+,\cdot).
Ring Theorynumber theory