MathDB

9

Part of 2000 AIME Problems

Problems(2)

System of Equations: Logs

Source:

1/5/2007
The system of equations \begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*} has two solutions (x1,y1,z1) (x_{1},y_{1},z_{1}) and (x2,y2,z2). (x_{2},y_{2},z_{2}). Find y1+y2. y_{1} + y_{2}.
logarithmsAMCAIMEalgebrasystem of equationsAIME Ibinomial theorem
Classic Complex Numbers

Source:

12/9/2005
Given that zz is a complex number such that z+1z=2cos⁡3∘,z+\frac 1z=2\cos 3^\circ, find the least integer that is greater than z2000+1z2000.z^{2000}+\frac 1{z^{2000}}.
trigonometryAMCAIME