Find the competition rankings of three people based on their final scores

I found this puzzle in my Logic and Reason book from 2000’s. The topic is ordering information. From the looks of it seems to be an adaptation from a reprinted version of Martin Gardner’s 70’s book of Puzzle Carnival.

The puzzle is as follows:

Marina, Sakura, and Hina were finalists at an Idol Athletics competition. They take part in the final, which has three trials: archery, rhythmic gymnastics and a 100-m sprint. In each test, the one who ends up first gets $a$ points, the second gets $b$ points and the third gets $c$ points. We know that $a$, $b$ and $c$ are positive integers such as $a>b>c$ there are no draws. In total Marina got 20 points, Sakura 10 points and Hina 9 points. We know that Marina ended up second place in the rhythmic gymnastics trial. Who ended up third in the archery trial and second place in 100-m sprint trial?

The choices given by my book are as follows:

1. Hina and Marina
2. Sakura and Hina
3. Marina and Sakura
4. Hina

I’m confused on how to arrange this information in a logical manner. My approach was to make a table. So far I have this table:

Where do I go from here?

It doesn’t say that $a$, $b$, and $c$ must be contiguous, but in order to have them to add up for 20, Marina’s first-place score, she must have ended up either third or first for either Archery or 100-m sprint. The same for the other two finalists, Hina and Sakura. How can this information be arranged more simply?

I attempted to break down the numbers to get 20, 10 and 9 and these are:

20 = 1+19, 2+18, 3+17, 4+16, 5+15, 6+14, 7+13, 8+12, 9+11, 10+10

But this didn’t help much. How can this puzzle be solved? Is there a trick or a method of simplification?

Should any sort of equation be used? Please include a diagram or sketch explaining how to approach this situation. Placing these people in order is very confusing for me, I don’t get what logic should be used.

The puzzle doesn’t specify the order the trials were conducted in. Would that affect the method of solution or does it not matter?