mateforum.ro

Azerbaijan didn't compete, although they were invited. Well done to Romania (once more)!

For the easiness of writing, let x=\lfloor \sqrt{n} \rfloor n=x^2+y,y<2x x+1\mid (x+1)(x-1)+y\rightarrow x+1\mid y x-1\mid (x+1)(x-1)+y+2\rightarrow x-1\mid y+2 1) y=0\rightarrow x\in {2,3}\rightarrow n\in {4,9} 2) y=x+1\rightarrow x-1\mid x+3\rightarrow x-1\mid 4\rightarrow x\in{2,3,5}\rightarrow n...

By Pigeonhole, there are 13 points of one color (50:4=12.5). From 13 points, there are \frac{13\cdot12}{2}=78 lines. From each line, we can make 2 isosceles triangles such that the given line is its base (if there are 3, those third points would lie on a line, contradiction). So we can make 78\cdot2...

\( (3,0,0)\succ (2,\frac{1}{2},\frac{1}{2}) \), by Muirhead it's true.

Another way: AM-GM: \( \frac{4a^3+b^3+c^3}{6}\ge\sqrt{a^4bc} \)

\( 4a^3+b^3+c^3\ge6a^2\sqrt{bc} \)
\( 4b^3+a^3+c^3\ge6b^2\sqrt{ac} \)
\( 4c^3+a^3+b^3\ge6c^2\sqrt{ab} \)
Sum, and that's it!

Are these Romanian Junior Balkan Team Selection Tests?

Can you translate problems to English?