practice problems

Q1. If b1,b2 ... .bn is a permutation of n positive real numbers a1,a2...an then find the minimum value of a1/b1 + a2/b2 ... an/bn.
S1.
Apply AM GM inequality to get answer = n.

Q2. If a1 + a2 ... an = 1, ai > 0 for all i then find the minimum value of 1/a1 + 1/a2 ... 1/an.
S2.
AM >= GM >= HM
HM = n/[1/a1 + 1/a2 .. 1/an] <= (a1 + a2 .. an)/n
=> n/X <= 1/n => X >= n^2

Q3. A,B are the A.M. and G.M. of two positive numbers a,b.
Show that, B < (a-b)^2/8(A-B) < A
S3.
(a-b)^2 = (a+b)^2 - 4ab = (2A)^2 - (2B)^2 = 4(A^2 - B^2)
So middle part becomes:
4(A-B)(A+B)/8(A-B) = (A+B)/2
B <= (A+B)/2 => B/2 <= A/2 which is correct. Similarly the other part.

Q4. a,b,c,d are distinct positive real numbers in H.P. Prove that:
1) a + d > b + c
2) ad > bc
S4.
let
1/a = A-3D
1/b = A-D
1/c = A+D
1/d = A+3D

1/a * 1/d = A^2 - 9D^2
1/b * 1/c = A^2 - D^2
Clearly 1/ad < 1/bc since D^2 is positive and A^2 is also positive.
=> ad > bc

Now:
1/a + 1/d = 1/b + 1/c
=> (a+d)/ad = (b+c)/bc
=> (a+d)/(b+c) = ad/bc > 1 => a+d > b+c
H.P.

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