I have next problem:
Can we using all parts from picture (every part exactly one time) to make rectangle?
I was thinking like: we have $20$ small square, so we have three possibility: $1 \times 20$, $2 \times 10$ and $4 \times 5$. I can see clearly that $1 \times 20$ and $2 \times 10$ are not possible. And some mine intuition says that we also can't make $4 \times 5$, but I can't prove it rigorous. Any help?
For the 4 by 5, suppose 4 rows and 5 columns, and consider rows 1,3 as blue and columns 1,3,5 as red. Then there are 10 blues and 12 reds. Now except for the T and L shapes, the other three contribute even numbers to either blue or red rows/columns. The L contributes an odd number to either blue or red, and the T contributes, depending on its orientation, either odd number to blue and an even number to red, or else an even number to blue and an odd number to red. So no matter how the T is oriented, we get an odd number for either the blues or the reds, which is impossible.
Easier proof: Take the 4 by 5 (or the 2 by 10) board and color it with black and white squares as in a traditional chess board. Then all but the T piece are such that, no matter where they are placed, they cover two black and two white squares. But the T shape must cover either 3 black and 1 white, or else the reverse 1 black and 3 white. So together the tiles cover either 9 black and 11 white, or else 11 black and 9 white. However both the 4 by 5 and the 2 by 10 board have 10 each of black and white.