MathDB

7

Part of 2016 AIME Problems

Problems(2)

Complex is Real

Source: 2016 AIME I #7

3/4/2016
For integers aa and bb consider the complex number ab+2016ab+100−(∣a+b∣ab+100)i.\dfrac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i. Find the number of ordered pairs of integers (a,b)(a, b) such that this complex number is a real number.
2016 AIME IAIMEcomplex numbers
Be There or be Square

Source: 2016 AIME II #7

3/17/2016
Squares ABCDABCD and EFGHEFGH have a common center and AB‾∥EF‾\overline{AB}\parallel \overline{EF}. The area of ABCDABCD is 20162016, and the area of EFGHEFGH is a smaller positive integer. Square IJKLIJKL is constructed so that each of its vertices lies on a side of ABCDABCD and each vertex of EFGHEFGH lies on a side of IJKLIJKL. Find the difference between the largest and smallest possible integer values of the area of IJKLIJKL.
AMCAIMEAIME II