"A teacher wrote down three positive real numbers on the blackboard and told Dima to decrease one of them by 3%, decreases another by 4% and increase the last by 5%. Dima wrote down the results in his notebook. It turned out that he wrote down the same three numbers that are on the blackboard, just in different order. Prove that Dima must have made a mistake"
"Just in a different order" I'm not clear with this line what I think they want to say that the order of the digits of number changed, I also found the solution here Q7 https://www.math.cmu.edu/~cargue/arml/archive/15-16/proofs-02-28-16-solutions.pdf I am getting no idea how can I prove that after doing what the teacher said to the order of digits of number must not have changed Any idea or hint would also be a great help
Edit : I solved the problem but I'm not Deleteing it because this might be helpfull for someone else in future
Explaining the Question :
Let us say that the teacher wrote $100,200,300$.
Dima could calculate like $97,192,315$
where $97=100(-3\%),192=200(-4\%),315=300(+5\%)$
Dima could also calculate $96,194,315$
where $96=100(-4\%),194=200(-3\%),315=300(+5\%)$
Dima could also calculate $96,210,291$
where $96=100(-4\%),210=200(+5\%),291=300(-3\%)$
There are other ways too.
In no such way will Dima write $100,200,300$ in that order or some other order.
Hence the teacher did not write $100,200,300$.
Now , let us say the teacher wrote $x,y,z$
Dima makes the calculations & will write $a,b,c$
It turns out that $a,b,c$ is actually $x,y,z$ in some other order !
We have to get the values of $x,y,z$.
Can you take it from here ?
You can try various things to figure it out.
Hints towards the Solution :
Let $x$ be the smallest.
Let $z$ be the largest.
Explaining the Answer :
When $x$ is smallest & it is reduced , it will become smaller , hence we want it to increase.
When $z$ is largest & it is increased , it will become larger , hence we want it to reduce.
If $x$ increase to $z$ & $z$ reduces to $x$ , then $y$ will be left out.
Hence the Possibility is to make $x$ increase to $z$ & $z$ reduce to $y$ & $y$ reduce to $x$.
Possibility 1 :
$x \times (1 + 0.05) = z$
$z \times (1 - 0.04) = y$
$y \times (1 - 0.03) = x$
Possibility 2 :
$x \times (1 + 0.05) = z$
$z \times (1 - 0.03) = y$
$y \times (1 - 0.04) = x$
Solve these to get 2 Solutions which may turn out unique.
That will let us Solve the actual Question asked.
Proceeding to Complete the Answer :
In Possibility 1 , we can substitute Second Equation $y$ into last Equation $y$ , then substitute that $x$ into first Equation to get :
$z \times (1 - 0.04) \times (1 - 0.03) = x$
$x \times (1 + 0.05) \times (1 - 0.04) \times (1 - 0.03) = x$
Hence $x=0$ , which gives $y=0$ & $z=0$
In Possibility 2 , we can substitute Second Equation $y$ into last Equation $y$ , then substitute that $x$ into first Equation to get :
$z \times (1 - 0.03) \times (1 - 0.04) = x$
$x \times (1 + 0.05) \times (1 - 0.03) \times (1 - 0.04) = x$
Hence $x=0$ , which gives $y=0$ & $z=0$
In no way we get Positive Solutions , yet the teacher wrote Positive numbers on the Black Board.
It is not Possible to get back the same Positive numbers in the same order or in some other order.
Hence Dima must have made a mistake in the calculations.
It may be like this :
When the teacher wrote $1003 , 1004 , 1005$ , then Dima wrongly increased $1003$ to $1005$ , wrongly reduced $1005$ to $1004$ , wrongly reduced $1004$ to $1003$ , getting back the original numbers via the great mistakes !
We can never know what the original numbers were & what mistakes Dima made !
We know that Dima made mistakes !