MathDB

2

Part of 1989 Iran MO (2nd round)

Problems(2)

Plane passes through a fixed point [Iran Second Round 1989]

Source:

12/6/2010
A sphere SS with center OO and radius RR is given. Let PP be a fixed point on this sphere. Points A,B,CA,B,C move on the sphere SS such that we have ∠APB=∠BPC=∠CPA=90∘.\angle APB = \angle BPC = \angle CPA = 90^\circ. Prove that the plane of triangle ABCABC passes through a fixed point.
geometry3D geometryspheregeometry proposed
Root of a polynomial

Source:

11/28/2010
Let nn be a positive integer. Prove that the polynomial P(x)=xnn!+xn−1(n−1)!+...+x+1P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 Does not have any rational root.
algebrapolynomialalgebra unsolved