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...