Find all functions \(f (x)\) such that \(f (2x + 1) = 4x^2 + 14x + 7\).
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 row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.