Olympiad problems

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: value , algebra

Find all possible values of $$\frac{(a+b-c)^2}{(a-c)(b-c)}+\frac{(b+c-a)^2}{(b-a)(c-a)}+\frac{(c+a-b)^2}{(c-b)(a-b)}$$

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: value , identity , algebra

Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$ where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$

2012 Bosnia and Herzegovina Junior BMO TST, 3

Tags: value , angle , geometry

Internal angles of triangle are $(5x+3y)^{\circ}$, $(3x+20)^{\circ}$ and $(10y+30)^{\circ}$ where $x$ and $y$ are positive integers. Which values can $x+y$ get ?

2015 Junior Regional Olympiad - FBH, 2

Tags: value

Show tha value $$A=\frac{(b-c)^2}{(a-b)(a-c)}+\frac{(c-a)^2}{(b-c)(b-a)}+\frac{(a-b)^2}{(c-a)(c-b)}$$ does not depend on values of $a$, $b$ and $c$

2025 Kyiv City MO Round 1, Problem 4

Tags: algebra , algebraic manipulations , value

Distinct real numbers $ a, b, c $ satisfy the following condition: \[ \frac{a - b}{a^3b^3} + \frac{b - c}{b^3c^3} + \frac{c - a}{c^3a^3} = 0. \] What are the possible values of the expression \[ \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}? \] [i]Proposed by Vadym Solomka[/i]