I am trying to solve Exercise 2.2.2 of "An Introduction to the Kolmogorov Complexity and Its Applications" (Li & Vitányi, Vol 3). The exercise is the following (paraphrased):

Let $ x $ satisfied $ K (x) geq n – O (1) $, or $ n = ell (x) $ is the length of $ x $ in a binary encoding. CA watch $ K (y) geq frac {n} {2} – O (1) $ for $ x = yz $ with $ ell (y) = ell (z) $.

I have tried but I have failed to apply the method of incompressibility, exploiting the fact that there are many $ x $ in a way. More direct approaches to trying to find intelligent recursive functions of $ y $ and $ z $ also did not work.