MathDB

1

Part of 2005 Kazakhstan National Olympiad

Problems(2)

Solve the half-trigonometric equation

Source: Kazakhstan NMO 2005 (Final round) P5

12/16/2014
Solve equation 212−2∣x∣=∣tan⁡x+12∣+∣tan⁡x−12∣2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|
trigonometryalgebra proposedalgebra
system of equations

Source: Kazakhstan NMO 2005 (Final round) P1

12/16/2014
Does there exist a solution in real numbers of the system of equations {(x−y)(z−t)(z−x)(z−t)2=A,(y−z)(t−x)(t−y)(x−z)2=B,(x−z)(y−t)(z−t)(y−z)2=C,\left\{ \begin{array}{rcl} (x - y)(z - t)(z - x)(z - t)^2 = A, \\ (y - z)(t - x)(t - y)(x - z)^2 = B,\\ (x - z)(y - t)(z - t)(y - z)^2 = C,\\ \end{array} \right. when a) A=2,B=8,C=6;A=2, B=8, C=6;
b) A=2,B=6,C=8.A=2, B=6, C=8.?
algebrasystem of equationsalgebra proposed