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.
For which natural \(n\) does the number \(\frac{n^2}{1.001^n}\) reach its maximum value?
Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?
We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the \(k\)-th word is obtained from the \((k-1)\)-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB...
a) Where in this sequence will the 1000th letter “A” be?
b) Prove that this sequence is non-periodic.
What figure should I put in place of the “?” in the number \(888 \dots 88\,?\,99 \dots 999\) (eights and nines are written 50 times each) so that it is divisible by 7?
Solve the equation \(x + \frac{1}{(y + 1/z)}= 10/7\) in natural numbers.
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.
Sage thought of the sum of some three natural numbers, and the Patricia thought about their product.
“If I knew,” said Sage, “that your number is greater than mine, then I would immediately name the three numbers that are needed.”
“My number is smaller than yours,” Patricia answered, “and the numbers you want are ..., ... and ....”
What numbers did Patricia name?
A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number. The result was three times greater.
Find these numbers.