Calculate the following squares in the shortest possible way (without
a calculator or any other device):
a) \(1001^2\) b) \(9998^2\) c) \(20003^2\) d) \(497^2\)
Real numbers \(x,y\) are such that \(x^2 +x \le y\). Show that \(y^2 +y \ge x\).
Today we will solve some problems using algebraic tricks, mostly
related to turning a sum into a product or using an identity involving
squares.
The ones particularly useful are: \((a+b)^2 =
a^2 +b^2 +2ab\), \((a-b)^2 = a^2 +b^2
-2ab\) and \((a-b) \times (a+b) = a^2
-b^2\). While we are at squares, it is also worth noting that any
square of a real number is never a negative number.
The evil warlock found a mathematics exercise book and replaced all the decimal numbers with the letters of the alphabet. The elves in his kingdom only know that different letters correspond to different digits \(\{0,1,2,3,4,5,6,7,8,9\}\) and the same letters correspond to the same digits. Help the elves to restore the exercise book to study.
The perimeter of the triangle \(ABC\) is 10. Let \(D,E,F\) be the midpoint of the segments \(AB,BC,AC\) respectively. What is the perimeter of the triangle \(DEF\)?