Excel in math, science, and engineering

Which of the above will not form a knot if the two ends of the string are pulled in opposite directions?

For how many values of \( \alpha \) does there exist a non-constant function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that

\[ f\left( \alpha (x + y ) \right) = f(x) + f(y) \]

Hello INMO, USAMO and all national olympiads awardees, try these problems and post solutions , specially for IMO 2016:

\((1)\) Find all functions \(f, g : R \rightarrow R\) which satisfy the ...

\[\large \dfrac{\sqrt{x}}{x^2+1+2\sqrt{xy}}+\dfrac{\sqrt{y}}{y^2+1+2\sqrt{2y-xy}}+\dfrac{\sqrt{2-x}}{x^2-4x+5+2\sqrt{2x-x^2}}\]

If ...

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