At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.
A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way (see the figure).
a) \(k\) \(L\)-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.
b) \(7\) \(G\)-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.