It seems that the total length of new set of $$$a$$$ or $$$b$$$ $$$\not= x$$$. Did I misunderstand something?

In my view, math education means you know a lot of math theories, such as linear algebra. But CP problems are more concrete and special. Does a player with good math education can do much better than a player who hasn't learn much math while having good thinking. (If we don't take account of the time they saved in learning prior knowledge).

Is the theory or the thinking that they trained leading success?

I don't think $$$f(k)=C(n,k)(n-1)^{n-k}$$$, because there should be only one k satisfying $$$cnt(k)=k$$$. For instance, $$$2,2,3,3,3$$$ is not 3-good array because we have both $$$cnt(3)=3$$$ and $$$cnt(2)=2$$$.

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I have had the feeling of the cycle. I didn't want to move or walk out of home, and my weight rapidly increased when I was depressed because eating food may be the easiest way to make me more comfortable. The key to breaking the cycle, in my experience, is sunshine. First thing when I get up is to get enough sunshine and then keep my life schedule consistent with the sun. If you participate in CF contests frequently and you live in Indonesia which I knew by your profile, staying up late might be your habit, which will worsen your depression.