$\begin{aligned} & \text { [ Arithmetic. Mental calculation, etc. } \\ & \text { [ Problems with inequalities. Case analysis ] }\end{aligned}$

In the olympiad, 2006 schoolchildren participated. It turned out that Vasya, a schoolboy, solved only one out of six problems, and the number of participants who solved

at least 1 problem is 4 times greater than those who solved at least 2;

at least 2 problems is 4 times greater than those who solved at least 3;

at least 3 problems is 4 times greater than those who solved at least 4;

at least 4 problems is 4 times greater than those who solved at least 5;

at least 5 problems is 4 times greater than those who solved all 6.

How many schoolchildren did not solve any problems?
The number of schoolchildren who solved at least one problem is greater than zero (since among them is Vasya), less than 2006, and divisible by \(4^5 = 1024\). Therefore, it is 1024, and the number of schoolchildren who did not solve any problems is \(2006 - 1024 = 982\).

## Answer

982 schoolchildren.

Task