MathDB

2

Part of 2007 Tuymaada Olympiad

Problems(2)

two polynomials of degree 100

Source: Tuymaada 2007, Problem 2

7/15/2007
Two polynomials f(x)=a100x100+a99x99+⋯+a1x+a0 f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0} and g(x)=b100x100+b99x99+⋯+b1x+b0 g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0} of degree 100 100 differ from each other by a permutation of coefficients. It is known that ai≠bi a_{i}\ne b_{i} for i=0,1,2,…,100 i=0, 1, 2, \dots, 100. Is it possible that f(x)≥g(x) f(x)\geq g(x) for all real x x?
algebrapolynomialalgebra proposed
slicing problem with a right angle in the end

Source: Tuymaada 2007, Problem 6

7/15/2007
Point D D is chosen on the side AB AB of triangle ABC ABC. Point L L inside the triangle ABC ABC is such that BD=LD BD=LD and ∠LAB=∠LCA=∠DCB \angle LAB=\angle LCA=\angle DCB. It is known that ∠ALD+∠ABC=180∘ \angle ALD+\angle ABC=180^\circ. Prove that ∠BLC=90∘ \angle BLC=90^\circ.
geometryperpendicular bisectorgeometry proposed