Question about linear combination of non-stationary signals
No, try with $\alpha_j = 0$, $\forall j$.

I suspect this to be a tricky question, to deceive fast thinking. For instance, having the "same length" does not mean a lot, for a continuous variable $t$. Misleading fake clues to make the question look serious.

Without more hypotheses on the notion of stationarity (in law, wide-sense) and the statistical interrelations between the so-called signals (it would be wiser to talk about processes). See Marcus' answer for more examples.