MathDB

3

Part of 2001 Poland - Second Round

Problems(2)

Monic polynomial with two high zero coefficients

Source: Polish Second Round 2001

3/6/2012
Let n≥3n\ge 3 be a positive integer. Prove that a polynomial of the form xn+an−3xn−3+an−4xn−4+…+a1x+a0,x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0, where at least one of the real coefficients a0,a1,…,an−3a_0,a_1,\ldots ,a_{n-3} is nonzero, cannot have all real roots.
algebrapolynomialalgebra proposed
Subsets with either odd and even sums of elements

Source: Polish Second Round 2001

3/6/2012
For a positive integer nn, let AnA_n and BnB_n be the families of nn-element subsets of Sn={1,2,…,2n}S_n=\{1,2,\ldots ,2n\} with respectively even and odd sums of elements. Compute ∣An∣−∣Bn∣|A_n|-|B_n|.
combinatorics proposedcombinatorics