MathDB

8

Part of 2019 AIME Problems

Problems(2)

trigonometric extrapolation

Source: 2019 AIME I #8

3/14/2019
Let xx be a real number such that sin⁡10x+cos⁡10x=1136\sin^{10}x+\cos^{10} x = \tfrac{11}{36}. Then sin⁡12x+cos⁡12x=mn\sin^{12}x+\cos^{12} x = \tfrac{m}{n} where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME I
Complicated Functions

Source: 2019 AIME ii #8

3/22/2019
The polynomial f(z)=az2018+bz2017+cz2016f(z)=az^{2018}+bz^{2017}+cz^{2016} has real coefficients not exceeding 20192019, and f(1+3i2)=2015+20193if(\tfrac{1+\sqrt{3}i}{2})=2015+2019\sqrt{3}i. Find the remainder when f(1)f(1) is divided by 10001000.
AIMEAMCAIME IIfunction2019 AIME II2019 AMCPolynomials