Let $n_1, n_2 , n_3, ..., n_9$ be nine distinct positive integers such that $n_1<n_2<n_3<...<n_9$ and $n_1+n_2+n_3+...+n_9 = 180.$ Suppose that the value of $n_1+n_2+n_3+n_4+n_5$ is maximum, find the maximum possible value of $n_9-n_1$.
The problem has thrown me off quite a bit because I don't understand how you can have $n_1+n_2+n_3+n_4+n_5$, the sum of smallest 5 numbers, be a maximum when as $i$ of $n_i$ increases the value of $n_i$ increases as well. If someone can help me understand the problem (and how to approach it) that'd be great.
Let's say you have written on a paper all the $9$-number sequences that solve the problem. For every sequence you can sum the $5$ smallest numbers to get the $n_1+n_2+n_3+n_4+n_5$ related to that sequence. For some sequences this would be a great number, for some other it will be small (the smallest it can be is $15$), for some other, it will be the greatest possible, e.g. the maximum, which is what you are looking for.
As for how to solve the problem, consider that there is the sequence $$(16,17,18,19,20,21,22,23,24)$$
Now consider another sequence. It is obvious that for this other sequence $n_5\geq 20$, if not the value for $n_1+n_2+n_3+n_4+n_5$ would be certainly less than for the above one. But if $n_5=20+x$, then the value of $n_5+n_6+n_7+n_8+n_9$ would be at least $5x$ greater than that of the one above, so the value for $n_1+n_2+n_3+n_4$ would be at least $5x$ smaller and $n_1+n_2+n_3+n_4+n_5$ would be $4x$ smaller.