That's a great question! The score system is based off your top 3 scores. This means your top 3 scores throughout the week are added together which will give you a final score. The reason why your score is not changing, is because you have not beaten any of your current top 3 scores.
Example: If you have 3 scores of 500, your total score = 1500. If you play again and get a score of 550, your new score would be 500+500+550 = 1550.
As for your question about competitions. Yes, there is a competition every week and a weekly prize!
If you do have any questions, you can also check out the website which may answer some of your questions.
Trivia Rush | Trivia Game App
Hope this helps!