MathDB

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Part of 2016 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2016 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
Find minimal value of A=(x+1x)6−(x6+1x6)−2(x+1x)3+(x3+1x3)A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}
algebraminimumInequality
Regional Olympiad - FBH 2016 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
If ∣ax2+bx+c∣≤1\mid ax^2+bx+c \mid \leq 1 for all x∈[−1,1]x \in [-1,1] prove that: a)a) ∣c∣≤1\mid c \mid \leq 1 b)b) ∣a+c∣≤1\mid a+c \mid \leq 1 c)c) a2+b2+c2≤5a^2+b^2+c^2 \leq 5
algebrainequalities
Regional Olympiad - FBH 2016 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
Let a1=1a_1=1 and an+1=an+12ana_{n+1}=a_{n}+\frac{1}{2a_n} for n≥1n \geq 1. Prove that a)a) n≤an2<n+n3n \leq a_n^2 < n + \sqrt[3]{n} b)b) lim⁡n→∞(an−n)=0\lim_{n\to\infty} (a_n-\sqrt{n})=0
limitalgebraSequence
Regional Olympiad - FBH 2016 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
Let aa and bb be real numbers bigger than 11. Find maximal value of c∈Rc \in \mathbb{R} such that 13+log⁡ab+13+log⁡ba≥c\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c
algebrainequalitiesmaximum value