11.1

Part of VMEO III 2006

Problems(2)

Calculate $P(\alpha^2+\alpha+1)$

Source: VMEO III

Given a polynomial

P(x)=x^4+x^3+3x^2-6x+1

P(\alpha^2+\alpha+1)

\alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}}

algebrapolynomial

11 contestants solve 9 math problems

Source: 2006 VMEO III Seniors 11.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

In a contest, there are

11

contestants to solve

9

math problems. After the end of the contest, it was found that any two contestants solved no more than

1

problem together. Find the largest positive integer

k

such that each problem can be solved by at least

k