MathDB

9

Part of 2012 AIME Problems

Problems(2)

Logarithms in x,y,z

Source: 2012 AIME I Problem 9

3/16/2012
Let xx, yy, and zz be positive real numbers that satisfy 2log⁡x(2y)=2log⁡2x(4z)=log⁡2x4(8yz)≠0. 2\log_x(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \neq 0. The value of xy5zxy^5z can be expressed in the form 12p/q\frac{1}{2^{p/q}}, where pp and qq are relatively prime integers. Find p+qp+q.
logarithmsnumber theoryrelatively primeAMC
Double Trigonometric Ratios

Source: 2012 AIME II Problem 9

3/29/2012
Let xx and yy be real numbers such that sin⁡xsin⁡y=3\frac{\sin{x}}{\sin{y}} = 3 and cos⁡xcos⁡y=12\frac{\cos{x}}{\cos{y}} = \frac{1}{2}. The value of sin⁡2xsin⁡2y+cos⁡2xcos⁡2y\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}} can be expressed in the form pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp + q.
ratiotrigonometryfunctionquadraticsnumber theoryrelatively primeAMC