Bogdanov I.I.

A natural number $a>1$ was appended to itself to form the number $b$, which is divisible by $a^{2}$. Find all possible values of the number $b / a^{2}$.
If the number \(a\) is \(n\)-digit, then \(\frac{b}{a^{2}}=\frac{\left(10^{n}+1\right) a}{a^{2}}=\frac{10^{n}+1}{a}\). Clearly, \(a \neq 10^{n-1}\), so \(1<\frac{10^{n}+1}{a}<10\). The number \(10^{n}+1\) (and hence the quotient) is not divisible by 2, 3 (the sum of the digits is 2), or 5, so

## Answer

$$
\begin{aligned} & Problem \(\underline{65419}\) Topics: \({[\) Mathematical Logic (other) \(] } \\ & {[\) Mean Values \(] }\) Difficulty: 4- \\ & Grades: 9,10,11 \end{aligned}
$$

Author: Shapovalov A.V.

In the zoo, there were 200 parrots. One day they each made one statement. Starting from the second, all statements were "Among the previously made statements, more than 70% are false." How many false statements did the parrots make in total?

## Solution

Note that if the first statement was true, then the second will be false, and vice versa, if the first statement was false, then the second will be true. Since the truth of the next statement depends only on the number of false statements among the previous ones, the truth of all remaining statements does not depend on what the first statement was. Therefore, without loss of generality, we will assume that it was false and write down the truth of the first eleven statements:

$$
\text { F, T, F, F, T, F, F, T, F, F, F... }
$$

Note that among the first 10 statements, exactly seven are false, so the 11th statement will be false. The truth of the following statements will not change if we consider the percentage of false statements starting from the 11th. Indeed, if among some set of statements more than \(70\%\) are false, then after adding another seven false statements and three true ones, the total number of false statements will still be more than \(70\%\).

Similarly, for sets where fewer than \(70\%\) of the statements are false and exactly \(70\%\) are false. Thus, the next 10 statements will completely repeat the first 10, meaning that among the first 20 statements, exactly 14 are false. Repeating the reasoning again, we get that the truth of statements from 21st to 30th coincides with the truth of the corresponding statements from 1st to 10th. And so on. Eventually, we will get 140 false statements out of two hundred.

## Answer

140 statements.