You are mixing four magic potions, and you choose how much of each one to use. Let \(a\), \(b\), \(c\), and \(d\) be the amounts of the four potions you pour in, each chosen between \(0\) and \(1\) liter. The wizard tells you that the magic power of your mix is given by the formula \[a + b + c + d - ab - bc - cd - da.\] What is the largest magic power you can create?
In a numerical set of \(n\) numbers, one of the numbers is 0 and another is 1.
a) What is the smallest possible variance of such a set of numbers?
b) What should be the set of numbers for this?
The numbers \(a_1, a_2, \dots , a_{1985}\) are the numbers \(1, 2, \dots , 1985\) rearranged in some order. Each number \(a_k\) is multiplied by its number \(k\), and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than \(993^2\).
Prove that there is a number of the form
a) \(1989 \dots 19890 \dots 0\) (the number 1989 is repeated several times, and then there are a few zeros), which is divisible by 1988;
b) \(1988 \dots 1988\), which is divisible by 1989.