Math Is Fun Forum

a,b,c>0 . Prove the inequality :
1/[a.\sqrt{a+b}] + 1/[b.\sqrt{b+c}]+1/[c.\sqrt{c+a}]    >=  3/(\sqrt{2abc}

triangle ABC (AB<AC )  , AD is a bisector prove with all I, J on AD we have
angleJBI >angleJCI

A)
I is a point in triangle ABC such that angle IBA > angle ICA , angle IBC > angle ICB , BI cut AC at B' , CI cut AB at C' . PRove the : BB' < CC'

please help me i'm very thanks

and the second problem
B)

a,b,c>0 prove the equation have 1 root :
d^3 - d(ab+bc+ca) - 2abc =0 .

from the equation and x,y,z >0 . ax+by+cz=xyz . Prove the inequality

x+y+z >= 2/d . (\sqrt[(a+d)(b+d)(c+d)])

C)in assemble N ,every number is white or black so that white + black=black
prove : white + white = white