1No reply????
1) Prove that nn> 1.3.5.....(2n-3)(2n-1)
2) Let a, b, c be the sides of a triangle with perimeter 2.
Prove that a2+b2+c2+2abc<2
3) -5<x<11. What is the greatest value of (11-x)3(x+5)5
4) x,y,z are positive real numbers such that x3y2z4=7 Show that :
2x+5y+3z ≥ 9(528/128)1/9
5) a,b,c are positive integers. Prove that
(aa/a+b+c)(bb/a+b+c)(cc/a+b+c )≥1/3(a+b+c)
6) x+y+z=0 and if a,b,c form the sides of a triangle then show that
a2yz+b2zx+c2xy is not positive
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11 Answers
211) simply taking 1,3,5 ... (2n-1) and writing AM≥GM
4)AM≥GM with 2x/3,2x/3,2x/3,5y/2,5y/2,3z/4,3z/4,3z/4,3z/4
215) Applying AM≥GM with a times 1/a ,b times 1/b , c times 1/c
and taking reciprocal
213)in the given interval 11-x and x+5 both are postive
writing 11-x = a and x+5=b
we can find the max value of a3b5
where a+b = 16
36@Fermat ->
I think there is some error in ur second question...
since, perimeter of the triangle is 2
=> (a + b + c) = 2
so how can a^2 + b^2 + c^2 + 2abc < 2, since a>0,b>0,c>0 (as sides of triangle can never be -ve)??
341Let a=x+y; b = y+z; c=z+x. Then we have x+y+z=1, and the given expression equals
\sum (x+y)^2 +2 (x+y)(y+z)(z+x) = \sum (1-x)^2 +2 (1-x)(1-y))(1-z)
= 3 -\sum x + \sum x^2 + 2 (1 - \sum x + \sum xy - xyz)
= 1 + \sum x^2 + 2 \sum xy - 2xyz = 1+ (x+y+z)^2 - 2xyz = 2 - 2xyz<2
12) @rahulmishra
(a + b + c)/3 ≥ a1/3.b1/3.c1/3
Cubing...
(a+b+c)3/27 ≥ abc
8/27 >= abc
Sides can be lesser than 1....
341The last one is to prove that a^2yz+b^2zx+c^2xy<0 for a,b,c sides of a triangle and x+y+z=0
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