Replace the letters in the word \(TRANSPORTIROVKA\) by numbers (different letters correspond to different numbers, but the same letters correspond to identical numbers) so that the inequality \(T > R > A > N < P <O < R < T > I > R > O < V < K < A\).
Restore the numbers. Restore the digits in the following example by dividing as is shown in the image
Decipher the numerical puzzle system \[\left\{\begin{aligned} & MA \times MA = MIR \\ & AM \times AM = RIM \end{aligned}\right.\] (different letters correspond to different numbers, and identical letters correspond to the same numbers).
This problem is from Ancient Rome.
A rich senator died, leaving his wife pregnant. After the senator’s death it was found out that he left a property of 210 talents (an Ancient Roman currency) in his will as follows: “In the case of the birth of a son, give the boy two thirds of my property (i.e. 140 talents) and the other third (i.e. 70 talents) to the mother. In the case of the birth of a daughter, give the girl one third of my property (i.e. 70 talents) and the other two thirds (i.e. 140 talents) to the mother.”
The senator’s widow gave birth to twins: one boy and one girl. This possibility was not foreseen by the late senator. How can the property be divided between three inheritors so that it is as close as possible to the instructions of the will?
The tower clock chimes three times in 12 seconds. How long will six chimes last?
The smell of a flowering lavender plant diffuses through a radius of 20 m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.
In some country there are 101 cities, and some of them are connected by roads. However, every two cities are connected by exactly one path.
How many roads are there in this country?
Determine all integer solutions of the equation \(yk = x^2 + x\). Where \(k\) is an integer greater than \(1\).
Prove that for every natural number \(n > 1\) the equality: \[\lfloor n^{1 / 2}\rfloor + \lfloor n^{1/ 3}\rfloor + \dots + \lfloor n^{1 / n}\rfloor = \lfloor \log_{2}n\rfloor + \lfloor \log_{3}n\rfloor + \dots + \lfloor \log_{n}n\rfloor\] is satisfied.
\(a_1, a_2, a_3, \dots\) is an increasing sequence of natural numbers. It is known that \(a_{a_k} = 3k\) for any \(k\). Find a) \(a_{100}\); b) \(a_{2022}\).