MathDB

P2

Part of 2022 District Olympiad

Problems(4)

sum cyc 1/(1+x^3)=3/2

Source: Romanian District Olympiad 2022 - Grade 9 - Problem 2

3/27/2022
a)a) Prove that 2x3−3x2+1≥0, (∀)x≥0.2x^3-3x^2+1\geq 0,~(\forall)x\geq0. b)b) Let x,y,z≥0x,y,z\geq 0 such that 21+x3+21+y3+21+z3=3.\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3. Prove that 1−x1−x+x2+1−y1−y+y2+1−z1−z+z2≥0.\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.
Inequalitycyclic inequalityinequalitiesalgebra
Romania District MO 2022 Grade 10 P2

Source: Romania District MO 2022 Grade 10

3/28/2022
Let z1,z2z_1,z_2 and z3z_3 be complex numbers of modulus 1,1, such that ∣zi−zj∣≥2|z_i-z_j|\geq\sqrt{2} for all i≠j∈{1,2,3}.i\neq j\in\{1,2,3\}. Prove that ∣z1+z2∣+∣z2+z3∣+∣z3+z2∣≤3.|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.Mathematical Gazette
complex numbersromaniainequalities
Romania District MO 2022 Grade 11 P2

Source: Romania District MO 2022 Grade 11

3/27/2022
Let A,B∈M3(R)A,B\in\mathcal{M}_3(\mathbb{R}) de matrices such that A2+B2=O3.A^2+B^2=O_3. Prove that det⁡(aA+bB)=0\det(aA+bB)=0 for any real numbers aa and b.b.
matrixcollege contestsromania
Romania District MO 2022 Grade 12 P2

Source: Romania District MO 2022 Grade 12

3/27/2022
Let (G,⋅)(G,\cdot) be a group and H≠GH\neq G be a subgroup so that x2=y2x^2=y^2 for all x,y∈G∖H.x,y\in G\setminus H. Show that (H,⋅)(H,\cdot) is an Abelian group.
group theoryabelian groupromaniacollege contests