MathDB

4

Part of 2003 AIME Problems

Problems(2)

Logs and Trig

Source:

5/11/2006
Given that log⁡10sin⁡x+log⁡10cos⁡x=−1\log_{10} \sin x + \log_{10} \cos x = -1 and that log⁡10(sin⁡x+cos⁡x)=12(log⁡10n−1)\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1), find nn.
trigonometrylogarithms
Tetrahedron Volume Ratios

Source:

12/26/2006
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is m/nm/n, where mm and nn are relatively prime positive integers. Find m+n.m+n.
geometry3D geometrytetrahedronratiovectornumber theoryrelatively prime