MathDB

1

Part of 2004 Regional Olympiad - Republic of Srpska

Problems(3)

very easy

Source: RS2004

3/20/2005
Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares of two positive integers.
number theory proposednumber theory
equation

Source: RS2004

3/20/2005
Find all real solutions of the equation x−13+3x−13=x+13.\sqrt[3]{x-1}+\sqrt[3]{3x-1}=\sqrt[3]{x+1}.
algebra proposedalgebra
polynomial and sequence

Source: RS2004

3/20/2005
Define the sequence (an)n≥1(a_n)_{n\geq 1} by a1=1a_1=1, a2=pa_2=p and an+1=pan−an−1 for all n>1.a_{n+1}=pa_n-a_{n-1} \textrm { for all } n>1. Prove that for n>1n>1 the polynomial xn−anx+an−1x^n-a_nx+a_{n-1} is divisible by x2−px+1x^2-px+1. Using this result, solve the equation x4−56x+15=0.x^4-56x+15=0.
algebrapolynomialalgebra proposed