KVPY SA Exam 2016 Question Paper

Question : 5 of 80

Marks: +1, -0

Let

a_1,a_2,...,a_100

be non-zero real numbers such that

a_1+a_2+...+a_100=0.

Then

$∑↖{100}↙{i=1}a_i2^{a_i} > 0$ and $∑↖{100}↙{i=1}a_i2^{-a_i} < 0$
$∑↖{100}↙{i=1}a_i2^{a_i} ≥ 0$ and $∑↖{100}↙{i=1}a_i2^{-a_i} ≥ 0$
$∑↖{100}↙{i=1}a_i2^{a_i} ≤ 0$ and $∑↖{100}↙{i=1}a_i2^{-a_i} ≤ 0$
the sign of $∑↖{100}↙{i=1}a_i2^{a_i}$ or $∑↖{100}↙{i=1}a_i2^{-a_i}$ depends on the choice of $a_i$

Solution:

Note that for every real number

a_i

a_i . 2^{a_i} > a_i

and

a_i . 2^{-a_i} < a_i

therefore,

∑↖{100}↙{i=1}a_i . 2^{a_i} > ∑↖{100}↙{i=1}a_i

and

∑↖{100}↙{i=1}a_i . 2^{-a_i} > ∑↖{100}↙{i=1}a_i

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