Let \(P\) be the product of the roots of \(z^6+z^4+z^3+z^2+1=0\) that have positive imaginary part, and suppose that \(P=r(\cos \theta^\circ+i\sin \theta^\circ),\) where \(0<r\) and \(0\le \theta <360.\) Find \(\theta.\)
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Source: AIME 1996 I Q11
Elo Rating: 1516.00
Suppose that \(x,\) \(y,\) and \(z\) are three positive numbers that satisfy the equations \(xyz=1,\) \(x+\frac{1}{z}=5,\) and \(y+\frac{1}{x}=29.\) Then \(z+\frac{1}{y}=\frac{m}{n},\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n.\)
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Source: AIME 2000 I Q7
Elo Rating: 1500.00