MathDB

4

Part of 2010 IberoAmerican Olympiad For University Students

Problems(1)

No consecutive zero coefficients

Source: XIII Ibero American Olympiad For University Students

7/22/2011
Let p(x)=xn+an−1xn−1+⋯+a1x+a0p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 be a monic polynomial of degree n>2n>2, with real coefficients and all its roots real and different from zero. Prove that for all k=0,1,2,⋯ ,n−2k=0,1,2,\cdots,n-2, at least one of the coefficients ak,ak+1a_k,a_{k+1} is different from zero.
algebrapolynomialinductionstrong inductionalgebra proposed