Innate Inequality

Algebra Level 3

Let a 1 , a 2 , a 3 , … , a 173 a_1,a_2,a_3,\ldots,a_{173} be positive real numbers such that a 1 + a 2 + a 3 + … + a 173 = 174 a_1+a_2+a_3+\ldots+a_{173}=174 . Find the minimum possible value of

a 1 2 a 1 + a 2 + a 2 2 a 2 + a 3 + a 3 2 a 3 + a 4 + … + a 173 2 a 173 + a 1 . \frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\frac{a_3^2}{a_3+a_4}+\ldots+\frac{{a_{173}}^{2}}{a_{173}+a_1}.


The answer is 87.

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Satvik Golechha by Satvik Golechha , 21, India

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6 solutions

Jubayer Nirjhor
Oct 19, 2014

Titu's Lemma.

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Rindell Mabunga
Sep 28, 2014

I used Cauchy then I got 87 87

1 Helpful 0 Interesting 0 Brilliant 0 Confused

By Cauchy-Schwarz in Engel Form, we have

a 1 2 a 1 + a 2 + a 2 2 a 2 + a 3 + ⋯ + a 173 2 a 173 + a 1 ≥ ( a 1 + a 2 + ⋯ + a 173 ) 2 2 ( a 1 + a 2 + ⋯ + a 173 ) = 87 \dfrac{a_{1}^{2}}{a_1+a_2} +\dfrac{a_{2}^{2}}{a_2+a_3} + \cdots + \dfrac{a_{173}^{2}}{a_{173} + a_1} \geq \dfrac{(a_1 + a_2 + \cdots + a_{173})^{2}}{2(a_1 + a_2 + \cdots + a_{173})}=\boxed{87}

Sean Ty - 6 years, 8 months ago

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I used jensens and put a 4 in the numerator cuz I accidentally squared the 2 up top. Careless

Trevor Arashiro - 6 years, 7 months ago

Yeah. CS is the easiest way to do it :D.

Krishna Ar - 6 years, 8 months ago
Dheeraj Agarwal
Jan 10, 2015

we can also apply AM-HM inequality.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Atharva Sarage
Oct 20, 2014

MODIFIED CAUCHY Schwarz

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Kushal Patankar
Oct 20, 2014

Consider all terms equal for extreme limits

0 Helpful 0 Interesting 0 Brilliant 0 Confused

there are 173 positive real numbers and their sum is 174.So from here we can get the minimum value of 1st 172 numbers as 1 and the minimum value of last number as 2.. By doing so the minimum value of the given equation will be 87.33. Since they are asking for the minimum integer number so the answer is 87

0 Helpful 0 Interesting 0 Brilliant 1 Confused

Absolutely Wrong Solution.

Kushagra Sahni - 5 years, 7 months ago

Abbbssssoooolllluuuttttteeeellly wwwwwwwwwwwwrrrrrrrrrrrrrrrrrrrrrooooooooooooooooonnnnnnnnnnnnnngggggggggg

Mohammed Imran - 1 year, 2 months ago

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