## [問題]Prove ### [問題]Prove

1.prove that in triangle ABC..
a.if a^2+b^2=c^2, then angle C = 90 degree
b.if a^3+b^3=c^3, then angle C is less than 90 degree
2.Given that p, m are real number and p^3+m^3=2.Prove that p+m is less  than or equal to 2
3.Given that a,b,c,m,n,p are real number and ap+cm=2bn,ac is greater than b^2. Prove that mp-n^2 is less than or equal to 0.
4.Prove that the sum of the squares of any five consecutive natural numbers cannot be a prefect square.

1.a.
a2+b2-c2=0，ab≠0，(a2+b2-c2)/2ab=cosC=0
0<C<π，C=0.5π
b.cosC=(a2+b2-c2)/2ab=(ca2+cb2-c3)/2abc=(ca2+cb2-a3-b3)/2abc
=[a2(1-a/c)+b2(1-b/c)]/2ab>0
cosC>0，0<C<π，0<cosC<0.5π

danny 2. p^3+m^3=(p+m)(p^2-pm+m^2)=2
if p+m>2, then p^2-pm+m^2 must <1.
P^2+2pm+m^2>4,
1>p^2-pm+m^2>4-3pm=>pm>1
and p^2+m^2>=2pm => p^2-pm+m^2>=pm>1 -><-

4.
We assume those five numbers are a-2 a-1 a a+1 a+2
The sum of their squares are 5a^2+10
5a^2+10=2(mod4) (if a is an even number)
5a^2+10=3(mod4) (if a is an odd number).
But perfects squares =1or0(mod4) -><- 追求神乎其技,至高無上的數學境界!~  3.
If mp>p^2,then
acmp>b^2n^2
4(bn)^2=(ac+mp)^2=a^2c^2+m^2p^2+2acmp
So that, (ac)^2+(mp)^2<2b^2n^2.
but, (ac)^2+(mp)^2>=2acmb>2b^2n^2
-><- 追求神乎其技,至高無上的數學境界!~  