MathDB

P2

Part of 2022 Romania National Olympiad

Problems(4)

Romania NMO 2022 Grade 9 P2

Source: Romania National Olympiad 2022

4/21/2022
Let ABCABC be a right triangle with ∠A=90∘.\angle A=90^\circ. Let A′A' be the midpoint of BC,BC, MM be the midpoint of the height ADAD and PP be the intersection of BMBM and AA′.AA'. Prove that tan⁡∠PCB=sin⁡C⋅cos⁡C.\tan\angle PCB=\sin C\cdot\cos C.
Daniel Văcărețu
romaniageometrytrigonometry
Romania NMO 2022 Grade 11 P2

Source: Romania National Olympiad 2022

4/20/2022
Let F\mathcal{F} be the set of pairs of matrices (A,B)∈M2(Z)×M2(Z)(A,B)\in\mathcal{M}_2(\mathbb{Z})\times\mathcal{M}_2(\mathbb{Z}) for which there exists some positive integer kk and matrices C1,C2,…,Ck∈{A,B}C_1,C_2,\ldots, C_k\in\{A,B\} such that C1C2⋯Ck=O2.C_1C_2\cdots C_k=O_2. For each (A,B)∈F,(A,B)\in\mathcal{F}, let k(A,B)k(A,B) denote the minimal positive integer kk which satisfies the latter property.
[*]Let (A,B)∈F(A,B)\in\mathcal{F} with det⁡(A)=0,det⁡(B)≠0\det(A)=0,\det(B)\neq 0 and k(A,B)=p+2k(A,B)=p+2 for some p∈N∗.p\in\mathbb{N}^*. Show that ABpA=O2.AB^pA=O_2. [*]Prove that for any k≥3k\geq 3 there exists a pair (A,B)∈F(A,B)\in\mathcal{F} such that k(A,B)=k.k(A,B)=k. Bogdan Blaga
linear algebraMatricesromania
Romania NMO 2022 Grade 10 P2

Source: Romania National Olympiad 2022

4/21/2022
Let z1z_1 and z2z_2 be complex numbers. Prove that ∣z1+z2∣+∣z1−z2∣⩽∣z1∣+∣z2∣+max⁡{∣z1∣,∣z2∣}.|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.Vlad Cerbu and Sorin Rădulescu
complex numbersromaniainequalities
Romania NMO 2022 Grade 12 P2

Source: Romania National Olympiad 2022

4/20/2022
Determine all rings (A,+,⋅)(A,+,\cdot) such that x3∈{0,1}x^3\in\{0,1\} for any x∈A.x\in A.
Mihai Opincariu
Ring Theoryromania