Question

The sum of several positive real numbers equal 10,
and the sum of squares of these numbers is not less than 20.
Prove that the sum of cubes is at least 40

Answer 1

@eliassaab @oldrin.bataku

Answer 2

several positive
worst case
all of them = 1
so 10 number ( if we dint count 0 case )

Answer 3

hmm there is much worst case since its real not integeral hmm

Answer 4

ok let it be
\(\large a_0+a_1+a_2+...+a_n=10\)
\(\large a_0^2+a_1^2+a_2^2+...+a_n^2\ge 20\)

Answer 5

wanna prove
\(\large a_0^3+a_1^3+a_2^3+...+a_n^3\ge 40\)

Answer 6

0.001 + 0.0001 + 0.1 + sqrt(2) + ... = 10

Answer 7

it need not be integers right

Answer 8

i already count that case

Answer 9

okay

Answer 10

i was thinking of its enough to prove
\(\large a_na_n^2\ge 2a_n+a^2\)
it work only for \(\large a_n > 1 \) for <1 or =1 there is a problem
so what to do ?

Answer 11

humm so all the several numbers cant be all <=1 at the same time :-|

Answer 12

if there is such away its like devides the numbers into two groups a_n >1 and a_n <=1 and say 1_ not all of them are 1 2_ not all of them are <1 (couse both of them wont satisfy : the sum of squares of these numbers is not less than 20. ) so there would be two cases :-
case 1:-
\(\large a_0+a_1+a_2+...+a_n=10\)
s.t a's >1
case 2 :-
\(\large a_0+a_1+a_2+...+a_n+ g_0+g_1+g_2+...+g_n=10\)
s.t
a's>1 , and g's <1

Answer 13

case 1 easy to prove :D
\( \large a_0+a_1+a_2+...+a_n=10...*\)
\(\large a_0^2+a_1^2+a_2^2+...+a_n^2\ge 20....**\)
\(\large a_0a_0^2+a_1a_1^2+a_2a_2^2+...+a_na_n^2 ...****\)
or multiply * with ** ur gonna have lower bound of 200 ( it would be complicated )
so for ***
we need to show
\(\large a_0a^2\ge 2a_n+a_n^2\)
( prove it bye induction for \(a_n \ge 2\) it will work and since its true sentence )
then
\(\large a_0a_0^2+a_1a_1^2+a_2a_2^2+...+a_na_n^2 ...\)
\(\large 2(a_0+a_1+a_2+...+a_n)+(a_0^2+a_1^2+a_2^2+...+a_n^2)\ge 2(10)+20=40 \)
so we proved hat
\(a_0^3+a_1^3+a_2^3+...+a_n^3\ge \)
for a'2 >=2
xD more cases incomming

Answer 14

@Mashy take a look if u have time :D

Answer 15

O.o

Answer 16

nvm lolz
at least i got an idea :3