There are 100 boxes, numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The audience can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and, without reading the questions aloud, honestly answers all of them. What is the minimum number of notes that need to be sent to definitely find out where the prize is?
Since the order of the answers to his questions is unknown to the audience, he must make a mistake-free choice, knowing only the number of "no" answers. If $N$ notes are sent, the number of "no" answers heard can take any integer value from 0 to $N$, meaning there are $N+1$ possible outcomes. This number must determine the number of the prize box, so its value must be no less than the number of boxes, that is, $N \geq 99$.

Let's consider an example of a set of 99 notes that meets the condition. Suppose the audience member submits to the host notes with a standard phrase: "Is it true that the number of the prize box does not exceed the number K?" for all K from 1 to 99. Assume the prize is in the box with number $m$. Then the audience member will hear $m-1$ "no" answers. By adding 1 to the number of "no" answers heard, he can determine the number of the prize box.

## Answer

99 notes.

Author: Bogdanov I.I.

Several numbers are written on the board. It is known that the square of each written number is greater than the product of any two other written numbers. What is the maximum number of numbers that can be on the board?

## Solution

Suppose there are at least four numbers, and $a$ is the number with the smallest absolute value. Among the remaining numbers, at least two have the same sign. Let's denote them as $b$ and $c$; then $b c=|b c| \geq\left|a^{2}\right|=a^{2}$, which contradicts the condition.

An example of three numbers that satisfy the condition: $1, 2, -3$.

## Answer

3 numbers.