MathDB

2

Part of 2000 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2000 Problem 2

Source: Turkey NMO 2000 Problem 2

9/29/2011
Let define Pn(x)=xn−1+xn−2+xn−3+⋯+x+1P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1 for every positive integer nn. Prove that for every positive integer aa one can find a positive integer nn and polynomials R(x)R(x) and Q(x)Q(x) with integer coefficients such that Pn(x)=[1+ax+x2R(x)]Q(x).P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).
algebrapolynomialnumber theoryrelatively primealgebra proposed
Turkey NMO 2000 Problem 5, Circles Have a Common Point

Source: Turkey NMO 2000 Problem 5

9/29/2011
A positive real number aa and two rays wich intersect at point AA are given. Show that all the circles which pass through AA and intersect these rays at points BB and CC where ∣AB∣+∣AC∣=a|AB|+|AC|=a have a common point other than AA.
geometry proposedgeometry