MathDB

P4

Part of 2022 Romania National Olympiad

Problems(4)

Romania NMO 2022 Grade 9 P4

Source: Romania National Olympiad 2022

4/21/2022
Let a<b<c<da<b<c<d be positive integers which satisfy ad=bc.ad=bc. Prove that 2a+a+d<b+c+1.2a+\sqrt{a}+\sqrt{d}<b+c+1.
Marius Mînea
romaniaalgebrainequalities
Some kind of functional equation

Source: Science ON 2021 grade X/2

3/8/2021
Let XX be a set with n≥2n\ge 2 elements. Define P(X)\mathcal{P}(X) to be the set of all subsets of XX. Find the number of functions f:P(X)↦P(X)f:\mathcal{P}(X)\mapsto \mathcal{P}(X) such that ∣f(A)∩f(B)∣=∣A∩B∣|f(A)\cap f(B)|=|A\cap B| whenever AA and BB are two distinct subsets of XX.
(Sergiu Novac)
algebracombinatoricsfunctional equation
Romania NMO 2022 Grade 11 P4

Source: Romania National Olympiad 2022

4/20/2022
Let A,B∈Mn(C)A,B\in\mathcal{M}_n(\mathbb{C}) such that A2+B2=2AB.A^2+B^2=2AB. Prove that for any complex number xxdet⁡(A−xIn)=det⁡(B−xIn).\det(A-xI_n)=\det(B-xI_n).Mihai Opincariu and Vasile Pop
linear algebraMatricesromania
Romania NMO 2022 Grade 12 P4

Source: Romania National Olympiad 2022

4/20/2022
Let (R,+,⋅)(R,+,\cdot) be a ring with center Z={a∈R:ar=ra,∀r∈R}Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\} with the property that the group U=U(R)U=U(R) of its invertible elements is finite. Given that GG is the group of automorphisms of the additive group (R,+),(R,+), prove that ∣G∣≥∣U∣2∣Z∩U∣.|G|\geq\frac{|U|^2}{|Z\cap U|}.Dragoș Crișan
Ring Theorygroup theoryromania