Difference between revisions of "1978 IMO Problems/Problem 5"
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| − | + | ==Problem== | |
| + | Let <math>f</math> be an injective function from <math>{1,2,3,\ldots}</math> in itself. Prove that for any <math>n</math> we have: <math>\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.</math> | ||
| + | |||
| + | ==Solution== | ||
| + | We know that all the unknowns are integers, so the smallest one must greater or equal to 1. | ||
| + | |||
| + | Let me denote the permutations of <math>(k_1,k_2,...,k_n)</math> with <math>(y_1,y_2,...,y_n)=y_i (*)</math>. | ||
| + | |||
| + | From the rearrangement's inequality we know that <math>\text{Random Sum} \geq \text{Reversed Sum}</math>. | ||
| + | |||
| + | We will denote we permutations of <math>y_i</math> in this form <math>y_n \geq ...\geq y_1</math>. | ||
| + | |||
| + | So we have <math>\frac{k_1}{1^2}+\frac{k_2}{2^2}+...+\frac{k_n}{n^2} \geq \frac{y_1}{1^2}+ \frac{y_2}{2^2}+...+ \frac{y_n}{n^2} \geq 1+\frac{1}{2}+...+\frac{1}{n}</math>. | ||
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| + | Let's denote <math>\frac{y_1}{1^2}+ \frac{y_2}{2^2}+...+ \frac{y_n}{n^2}=T</math> and <math>1+\frac{1}{2}+...+\frac{1}{n}=S</math>. | ||
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| + | We have <math>T \geq S</math>. Which comes from <math>y_1 \geq1, y_2 \geq2, ...,y_n \geq n</math>. | ||
| + | |||
| + | So we are done. | ||
| + | |||
| + | The above solution was posted and copyrighted by Davron. The original thread for this problem can be found here: [https://aops.com/community/p509573] | ||
| + | |||
| + | == See Also == {{IMO box|year=1978|num-a=4|num-a=6}} | ||
Revision as of 16:05, 29 January 2021
Problem
Let 
be an injective function from 
in itself. Prove that for any 
we have: 
Solution
We know that all the unknowns are integers, so the smallest one must greater or equal to 1.
Let me denote the permutations of 
with 
.
From the rearrangement's inequality we know that 
.
We will denote we permutations of 
in this form 
.
So we have 
.
Let's denote 
and 
.
We have 
. Which comes from 
.
So we are done.
The above solution was posted and copyrighted by Davron. The original thread for this problem can be found here: [1]
See Also
| 1978 IMO (Problems) • Resources | ||
| Preceded by [[1978 IMO Problems/Problem {{{num-b}}}|Problem {{{num-b}}}]] |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
