If you want to favour finishing some problems over nibbling at every problem, you can simply change the way you "give partial points" to each problem to be more convex.
If we take the sale from @MadJack,
- no answer or very, very wrong,
- got started, made some progress, but took a wrong turn somewhere,
- half-way there,
- minor error,
Then instead of ranking those (for an easy example, out of 8) as 0, 2, 4, 6, 8, you could do 0, 1, 2, 4, 8.
This is exponential, so maybe an extreme example, but any convex function would do to shift the reward from equally attempting any problem to finishing problems.
Alternately you could reward the introductory "parts" with 1 point and the final ones with 3 points, so the grades would be 0, 1, 2, 5, 8.
To find the right function for you, it depends on your preferences (whether to force integer scores, how to weight different levels of completeness of the exercise, etc). I guess you should experiment a little maybe on the previous tests you had to see how it would modify the grade based on what you do, as well.
The main take-away is that the rewarded strategy for a convex scale is to finish exercises.
A different example of this is what one of my teachers used to do: grade 7 questions out of 5, starting from 5 and getting -1 per unanswered or wrongly answered question (until you reach 0, no negative grades).
If you look at it by adding points instead of subtracting them, the (cumulated) points that each successive (correctly) answered question grants are: 0, 0, 1, 2, 3, 4, 5, which is indeed convex. If this is applied to an exercise in a test, the answering strategy becomes to finish every exercise you start, but only attempt those where you feel you would get more than 2 questions right.
This specific scale is a little harsh however - so I wouldn't really recommend it - and was meant for questions on definitions and such that everybody was expected to know.