MathDB

1

Part of 1981 Vietnam National Olympiad

Problems(2)

Solving a system of four equations in four variables.

Source: Vietnam MO 1981 P4

3/17/2011
Solve the system of equations x2+y2+z2+t2=50;x^2 + y^2 + z^2 + t^2 = 50; x2−y2+z2−t2=−24;x^2 - y^2 + z^2 - t^2 = -24; xy=zt;xy = zt; x−y+z−t=0.x - y + z - t = 0.
algebrasystem of equationsalgebra unsolved
ABC is right triangled iff cyc sum sin A=1+cyc sum cos A

Source: Vietnam MO 1981 P1

3/17/2011
Prove that a triangle ABCABC is right-angled if and only if sin⁡A+sin⁡B+sin⁡C=cos⁡A+cos⁡B+cos⁡C+1\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1
trigonometrygeometryperimetercircumcircleinradiusgeometry unsolved