MathDB

P2

Part of 2023 District Olympiad

Problems(4)

Equation with floor function

Source: Romanian District Olympiad 2023 9.2

3/11/2023
[*]Determine all real numbers xx{} satisfying ⌊x⌋2−x=−0.99\lfloor x\rfloor^2-x=-0.99. [*]Prove that if a⩽−1a\leqslant -1, the equation ⌊x⌋2−x=a\lfloor x\rfloor^2-x=a does not have real solutions.
algebrafloor function
Several triangles are equilateral

Source: Romanian District Olympiad 2023 10.2

3/11/2023
Let ABCABC be an equilateral triangle. On the small arc ABAB{} of its circumcircle Ω\Omega, consider the point NN{} such that the small arc NBNB measures 30∘30^\circ{}. The perpendiculars from NN{} onto ACAC and ABAB intersect Ω\Omega again at PP{} and QQ{} respectively. Let H1,H2H_1,H_2 and H3H_3 be the orthocenters of the triangles NAB,QBCNAB, QBC and CAPCAP respectively.
[*]Prove that the triangle NPQNPQ is equilateral. [*]Prove that the triangle H1H2H3H_1H_2H_3 is equilateral.
geometrycomplex numbers
Inverses and determinants

Source: Romanian District Olympiad 2023 11.2

3/11/2023
Let AA{} and BB be invertible n×nn\times n matrices with real entries. Suppose that the inverse of A+B−1A+B^{-1} is A−1+BA^{-1}+B. Prove that det⁡(AB)=1\det(AB)=1. Does this property hold for 2×22\times 2 matrices with complex entries?
linear algebramatrixdeterminant
Group problem

Source: Romanian District Olympiad 2023 12.2

3/11/2023
Let (G,⋅)(G,\cdot) be a grup with neutral element ee{}, and let HH{} and KK be proper subgroups of GG, satisfying H∩K={e}H\cap K=\{e\}. It is known that (G∖(H∪K))∪{e}(G\setminus(H\cup K))\cup\{e\} is closed under the operation of GG. Prove that x2=ex^2=e for all the elements xx{} of GG{}.
abstract algebragroup theoryalgebra