MathDB

3.1

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

sin^3 a/ sin b +cos^3 a/cos b >= 2/cos(a - b)

Source: 2011 Saudi Arabia Pre-TST November 3.1

1/1/2022
Prove that sin⁡3asin⁡b+cos⁡3acos⁡b≥1cos⁡(a−b)\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)} for all aa and bb in the interval (0,π/2)(0, \pi/2) .
algebrainequalitiestrigonometry
2011^{2012} divides n! if 2011^{2011} divides n!

Source: 2011 Saudi Arabia Pre-TST February 3.1

1/1/2022
Let nn be a positive integer such that 201120112011^{2011} divides n!n!. Prove that 201120122011^{2012} divides n!n! .
number theorydividesdivisible