2 problems of Polish MO 2007

northwolves

1 Polynomial P(x) has integer coefficients. Prove, that if polynomials P(x) and P(P(P(x))) have common real root, they also have a common integer root. 2 a, b ,c ,d are positive integers and ad=b²+bc+c² Prove that a²+b²+c²+d² is a composed number.

mathe

第一题好像不难: 假设P(x)和P(P(P(x)))有公共解u 由P(P(P(u)))=0得 0=P(P(P(u)))=P(P(0)) 我们得到P(0)是函数P(x)的一个整数解 而P(P(P(P(0)))=P(P(0))=0 所以P(x)和P(P(P(x)))有公共整数解P(0).

mathe

(a+d+b+c)(a+d-b-c) =(a+d)^2-(b+c)^2 =a^2+d^2-b^2-c^2+2(ad-bc) =a^2+b^2+c^2+d^2 余下只要证明a+d-b-c>1 a+d>=2sqrt(ad)>(1/sqrt(3) + sqrt(2))sqrt(ad)=sqrt(bc+b^2+c^2)/sqrt(3)+sqrt(2)sqrt(b^2+c^2+bc) >=sqrt(1+1+1)/sqrt(3)+sqrt(2b^2+2c^2+2bc) >1+sqrt(b^2+c^2+2bc)=1+b+c 得证

数学星空

呵,mathe数学分析能力挺强的.... 是数学系的吧...