MathDB

2

Part of 1990 Greece National Olympiad

Problems(3)

a/(b^3-1)+b/(a^3-1)=2(ab-2)/ (a^2b^2+3) 1990 Greece MO Grade X p2

Source:

9/6/2024
If a+b=1a+b=1, ∈R \in \mathbb{R} and ab≠0ab \ne 0, prove that ab3−1+ba3−1=2(ab−2)a2b2+3\frac{a}{b^3-1}+\frac{b}{a^3-1}=\frac{2(ab-2)}{a^2b^2+3}
algebra
\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}

Source: 1990 Greece MO Grade XI p2

9/6/2024
Find all real solutions of x−1+x2−1=x3\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}
algebraRadicals
vectod OK x OL+OL x OM + OM x On + ON x OK = fixed, midpoints of square

Source: 1990 Greece MO Grade XII p2

9/6/2024
Let ACBDACBD be a asquare and K,L,M,NK,L,M,N be points of AB,BC,CD,DAAB,BC,CD,DA respectively. If OO is the center of the square , prove that the expression OK→⋅OL→+OL→⋅OM→+OM→⋅ON→+ON→⋅OK→ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK} is independent of positions of K,L,M,NK,L,M,N, (i.e. is constant )
vectorfixedgeometry