In a certain country $10\%$ of the employees get $90\%$ of the total salary paid in this country. Supposing that the country is divided in several regions, is it possible that in every region the total salary of any 10% of the employees is no greater than $11\%$ of the total salary paid in this region?
Problem
Source: ToT - 2001 Spring Junior A-Level #1
Tags: algebra unsolved, algebra
14.08.2014 21:38
Yes. It is enough to show example. Take $200$ employees and $2$ regions. From the condition we have $ 0.1 * 200 = 20 $ so $ 20$ employees together have $90 \%$ of the country salary . Take those $ 20 $ and put them in the first region and the other $180$ employees in the second. Let country salary be $200$ and let everyone in the first region have salary $9$. First region: $ 10 \%$ equals $2$ employees and sum of every two employee's salaries is $ 9 + 9 = 18 $ and that is $ \frac{18}{9*20} = 10\% < 11\% $ which is what we wanted. Second region : Everyone of $180$ employees has salary $\frac{20}{180} = \frac {1}{9}$ We have $ 10\% $ equals $ 18 $ employees and their salary sum is always $ 18* \frac{1}{9} = 2$ but $ \frac{2}{20} = 10\% < 11\% $ which is what we wanted.
15.08.2014 08:09
No, it is not enough to show an example. You only proved that the statement holds for the particular example you gave. You need to prove that the statement holds for all possibilities (or otherwise disprove with an example).
15.08.2014 08:53
The claim clearly does not hold in all cases. Say $99$ employees have a salary of $1$, and the $100$-th has a salary of $81$. Then the region this employee is in fails the test.
15.08.2014 10:56
Ok, thanks. I thought that it is enough to show the example of possibility because of " is it possible ? ".
15.08.2014 11:28
Oh, sorry. Actually yes, an example seems enough given the wording. It's not quite clear which variables we can control.