I noticed that when you solve a problem there isn’t anything indicating that, like for example a green tick image.
Is there any reason why this is the case?
Codechef does not automatically mark solved problems for you. They are already providing us with lots of good challenging problems to compete, that they have enough to do around, which is more important than making this feature.
But, like @kunal361 said, we can use extensions/add-ons for this purpose. There are many available.
You can read these posts (discussion #1, discussion #2) for more details and links.
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To be correct, now you can install it in firefox too 
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