What is a circle in the 101th dimension?
Given that the numbers are all positive reals x 1 2 + x 2 2 + ⋯ + x 1 0 1 2 = 1
Maximize n = 2 ∑ 1 0 1 1 7 2 9 x 1 x n .
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1 solution
Your "equity" case needs to be adjusted to the new situation!
Don't you have just 99 quantities 1 2 in the first row? My answer came out to be 2 1 7 2 9 9 9 , "none of the above"
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you are right. i will change the wording to 101 dimension and you can report the problem. i dont know why but all my problems have wrong answer at first....
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Just a small error, easy to correct... the problem reminds me of this one but you push it further in an interesting way (+1).
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@Otto Bretscher – yes it was inspired by those problems, but too many problems had the same idea so i didnt tag any.
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@Aareyan Manzoor – The theme here is the "common term", x 1 , which makes them easy to solve .
I solved it in this (analogous) way,
0 ≤ k = 2 ∑ 1 0 1 ( x 1 − 1 0 x k ) 2 = 1 0 0 k = 1 ∑ 1 0 1 x k 2 − 2 0 k = 2 ∑ 1 0 1 x 1 x k = 1 0 0 − 2 0 k = 2 ∑ 1 0 1 x 1 x k so that ∑ k = 2 1 0 1 x 1 x k ≤ 5 and ∑ k = 2 1 0 1 1 7 2 9 x 1 x k ≤ 8 6 4 5
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@Otto Bretscher – yes. but what would be the case without common term , i.e n = 1 ∑ 1 0 1 x n x n + 1 . interesting theme for my next problem. i will try to solve this one now.Help will be appreciated though.
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@Aareyan Manzoor – This one is trickier. There are two versions... you can link it up with x 1 0 1 x 1 or not. Enjoy!
Are you interested in my matrix problems?
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@Otto Bretscher – well, i havent learned linear algebra properly yet. i only know like the basic and lagrange multipliers for linear system i can solve. i probably cant solve any of them.....
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@Aareyan Manzoor – It's good stuff, well worth studying, easy for a guy like you. Get a copy of my text book! ;)
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@Otto Bretscher – this comment has just encouraged me to go and learn linear algebra! thanks for that,
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@Aareyan Manzoor – Linear algebra is a lot easier than Dirichlet series, and a lot more important ;)
@Aareyan Manzoor – Let me pose the easy version and I will leave the trickier one for you ;)
You could have also chosen AM-GM
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Lol I got the answer and then forgot To multiply by 1729.
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Nice beta JSTSE hum dono ka clear.
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@Department 8 – abhi tak to chalu nahi kiya. ab aage dekhte hai. tere ko kya tension teri to gk strong hai.
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@Shreyash Rai – Abe chal meri silly mistakes boht zyada hain
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@Department 8 – Silly mistakes Ki kahania suna dunga to..... Leave it we will talk about that somewhere else
write as 1 7 2 9 x 1 n = 2 ∑ 1 0 1 x n ≤ Cauchy-Swarz 1 7 2 9 x 1 ( 100 times 1 2 + 1 2 + . . + 1 2 ) ( x 2 2 + x 3 2 + . . . + x 1 0 1 2 ) the expression equals 1 7 2 9 x 1 ( 100 times 1 2 + 1 2 + . . + 1 2 ) ( x 2 2 + x 3 2 + . . . + x 1 0 1 2 ) = 1 7 2 9 1 0 0 x 1 2 ( 1 − x 1 2 ) = 1 7 2 9 0 x 1 2 ( 1 − x 1 2 ) since all are positive, 1 ≥ x 1 2 ⟹ 1 − x 1 2 ≥ 0 . so AM-GM is legal 1 7 2 9 0 x 1 2 ( 1 − x 1 2 ) ≤ AM-GM 1 7 2 9 0 2 x 1 + ( 1 − x 1 ) = 2 1 7 2 9 0 = 8 6 4 5
equity case x 1 = 2 1 , x 2 = x 3 = x 4 = . . = x 1 0 1 = 2 0 3