MathDB

6

Part of 2012 AIME Problems

Problems(2)

Imaginary part of a complex number

Source: 2012 AIME I Problem 6

3/16/2012
The complex numbers zz and ww satisfy z13=wz^{13} = w, w11=zw^{11} = z, and the imaginary part of zz is sin⁡(mπn)\sin\left(\frac{m\pi}n\right) for relatively prime positive integers mm and nn with m<nm < n. Find nn.
trigonometryAMCAIMEconicsparabolaLaTeXcomplex numbers
Distance Between Complex Numbers

Source: 2012 AIME II Problem 6

3/29/2012
Let z=a+biz = a + bi be the complex number with ∣z∣=5|z| = 5 and b>0b > 0 such that the distance between (1+2i)z3(1 + 2i)z^3 and z5z^5 is maximized, and let z4=c+diz^4 = c + di. Find c+dc+d.
inequalitiesgeometrygeometric transformationreflectiontriangle inequalityAMC