MathDB

3

Part of 2007 AIME Problems

Problems(2)

Complex Number Problem

Source: AIME I 2007 #3

3/15/2007
The complex number zz is equal to 9+bi9+bi, where bb is a positive real number and i2=āˆ’1i^{2}=-1. Given that the imaginary parts of z2z^{2} and z3z^{3} are equal, find bb.
AMC
Long Diagonal

Source: AIME II 2007 #3

3/29/2007
Square ABCDABCD has side length 1313, and points EE and FF are exterior to the square such that BE=DF=5BE=DF=5 and AE=CF=12AE=CF=12. Find EF2EF^{2}. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("AA", A, dir(point--A)); label("BB", B, dir(point--B)); label("CC", C, dir(point--C)); label("DD", D, dir(point--D)); label("EE", E, dir(point--E)); label("FF", F, dir(point--F));[/asy]
geometrycircumcirclesymmetryrectangleanalytic geometryrotationPythagorean Theorem