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Cardinality of Sets

Let A=a(1) a(...

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Let A={a(1) a(2) a(3) a(4)} where a(1)>a(2)>a(3)>a(4) The total number of unordered pairs of disjoint subsets of A is equal to

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a_{1}, a_{2}, a_{3}, a_{4}, a_{5} > 0

\frac{a_{1}}{a_{2} + a_{3}} + \frac{a_{2}}{a_{3} + a_{4}} + \frac{a_{3}}{a_{4} + a_{5}} + \frac{a_{4}}{a_{5} + a_{1}} + \frac{a_{5}}{a_{1} + a_{2}} \geq \frac{5}{2}

a_{1}, a_{2}, a_{3}, a_{4}

are in AP, then

\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \frac{1}{\sqrt{a_{3}} + \sqrt{a_{4}}}

Q. It is a KVPY question Let a1,a2,a3,a4 be real number such that a1+a2+a3+a4=0 and a1^2+a2^2+a3^2+a4^2=1 then the smallest possible value of the expression (a1-a2)^2+(a2-a3)^2+(a3-a4)^2+(a4-a1)^2 lies in which interval. Not sure about the chapter..

a_{1}, a_{2}, a_{3}, a_{4}, a_{5} a n d a_{6}

are consecutive odd natural numbers such that

a_{6} > a_{1}, a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} = p^{3} a n d a_{5} = 39

, then the value of

(a_{3} + a_{4} - 12 p)

a_{1}, a_{2}, a_{3}, a_{4}, . . {. ., a}_{20}

a_{1} + a_{20} = 45

a_{1} + a_{2} + a_{3} + a_{4}, . . {. . + a}_{20}

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