MHB Sum of 100 Terms: Prove At Least 2 Numbers Equal

[anemone]

The natural numbers $a_1,\,a_2,\,\cdots,\,a_{100}$ are such that

$\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_1}}+\cdots+\dfrac{1}{\sqrt{a_1}}=20$.

Prove that at least two of the numbers are equal.

 

[lfdahl]

Spoiler

Hi, anemone
I´d expected the sum to be: $\frac{1}{\sqrt{a_1}}+\frac{1}{\sqrt{a_2}}+...+\frac{1}{\sqrt{a_{100}}}$
- or am I wrong??

 

[anemone]

The natural numbers $a_1,\,a_2,\,\cdots,\,a_{100}$ are such that

$\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_1}}+\cdots+\dfrac{1}{\sqrt{a_1}}=20$.

Prove that at least two of the numbers are equal.

Ops...typo...again...sorry folks!

The problem should read:

The natural numbers $a_1,\,a_2,\,\cdots,\,a_{100}$ are such that

$\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_2}}+\cdots+\dfrac{1}{\sqrt{a_{100}}}=20$.

Prove that at least two of the numbers are equal.

 

[lfdahl]

A hint is requested

 

[anemone]

A hint is requested

Hello MHB!

Something irritating has happened to my laptop and it seems like the folder that contains all challenging problems that I have collected from all over the world is ... gone...(Sweating)(Worried)

I will post back for any update on my effort to save the situation and for this challenge problem, I am afraid I may need some decent time to look for its source so I could post a hint based on the suggested solution I found online...sorry folks!

 

[Albert1]

Ops...typo...again...sorry folks!

The problem should read:

The natural numbers $a_1,\,a_2,\,\cdots,\,a_{100}$ are such that

$\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_2}}+\cdots+\dfrac{1}{\sqrt{a_{100}}}=20$.

Prove that at least two of the numbers are equal.

Spoiler

let $a_1\neq a_2\neq a_3\neq ----------\neq a_{100}----(1)$
$$S=\dfrac {1}{\sqrt 1}+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{4}}+\dfrac{1}{\sqrt{5}}+----+\dfrac{1}{\sqrt{100}}
<1+\int_{1}^{100}\dfrac{dx}{\sqrt{x}}=1+18=19--(2)$$
but we are given $S=20---(3)$
a contradiction between (2) and (3)
(1) is impossible
so at least two of the numbers are equal

 

[anemone]

Spoiler

let $a_1\neq a_2\neq a_3\neq ----------\neq a_{100}----(1)$
$$S=\dfrac {1}{\sqrt 1}+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{4}}+\dfrac{1}{\sqrt{5}}+----+\dfrac{1}{\sqrt{100}}
<1+\int_{1}^{100}\dfrac{dx}{\sqrt{x}}=1+18=19--(2)$$
but we are given $S=20---(3)$
a contradiction between (2) and (3)
(1) is impossible
so at least two of the numbers are equal

Very well done Albert!(Cool)