MathDB

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Part of 2017 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2017 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
Let aa, bb and cc be real numbers such that abc(a+b)(b+c)(c+a)≠0abc(a+b)(b+c)(c+a)\neq0 and (a+b+c)(1a+1b+1c)=10071008(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008} Prove that ab(a+c)(b+c)+bc(b+a)(c+a)+ca(c+b)(a+b)=2017\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017
identityreal numbersalgebra
Regional Olympiad - FBH 2017 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
If aa is real number such that x1x_1 and x2x_2, x1≠x2x_1\neq x_2 , are real numbers and roots of equation x2−x+a=0x_2-x+a=0. Prove that ∣x12−x22∣=1\mid {x_1}^2-{x_2}^2 \mid =1 iff ∣x13−x23∣=1\mid {x_1}^3-{x_2}^3 \mid =1
equationalgebraroots
Regional Olympiad - FBH 2017 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
In terms of real parameter aa solve inequality: log⁡ax+∣a+log⁡ax∣⋅log⁡xa≥alog⁡xa\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a} in set of real numbers
parameterizationalgebralogarithmsinequalities