  In this paper we study the radial and orthogonal projections and the distance
sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called
Mandelbrot percolation or percolation fractals. We prove that the following
assertion holds almost surely: if the Hausdorff dimension of $E$ is greater
than 1 then the orthogonal projection to \textbf{every} line, the radial
projection with \textbf{every} center, and distance set from \textbf{every}
point contain intervals.
