MathDB

2

Part of 2015 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2015 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
Let aa, bb and cc be positive real numbers such that abc=2015abc=2015. Prove that a+ba2+b2+b+cb2+c2+c+ac2+a2≤a+b+c2015\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}
algebraInequality
Regional Olympiad - FBH 2015 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
Find all triplets (p,a,b)(p,a,b) of positive integers such that p=ba−8ba+8bp=b\sqrt{\frac{a-8b}{a+8b}} is prime
number theoryprime
Regional Olympiad - FBH 2015 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
Let aa, bb and cc be positive real numbers such that abc=1abc=1. Prove the inequality: 1a+b+1b+c+1c+a≤a2+b2+c22\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}
inequalitiesalgebra
Regional Olympiad - FBH 2015 Grade 12 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
For positive integer nn, find all pairs of coprime integers pp and qq such that p+q2=(n2+1)p2+qp+q^2=(n^2+1)p^2+q
number theorycoprime