1. Many young people are fond of... football and other kinds of sports.

C. playing is the correct answer. The phrase "fond of" is followed by a gerund (verb + -ing).
2. They couldn't help... when they heard the little boy singing a love song.

C. laugh is the correct answer. "Couldn't help + verb-ing" is a fixed expression meaning "unable to prevent oneself from doing something."
3. I remember... them to play in my garden.

A. to allow is the correct answer. The structure is "remember + to + verb" for something you need to do in the future.
4. It was a nasty memory. Do you remember both of us wearing sunglasses to avoid... by the supervisors?

D. being recognized is the correct answer. The phrase "avoid + gerund" is used to express the purpose of an action.
5. I can't bear thinking back of that time. I'd rather... equally.

B. be treated is the correct answer. "Would rather + verb" is used to express preference.
6. Did you accuse Nam of... with his elbow? Well, I saw him... it off the table.

B. breaking/knocking is the correct answer. The first blank requires the gerund form ("of + gerund") and the second blank requires the present participle form (used with the verb "saw").
7. We found it very difficult... with Gamma.

A. to work is the correct answer. The structure is "find + it + adjective + to + verb."
8. I can't read when I am traveling. It makes me... sick.

D. feeling is the correct answer. The structure is "make + object + verb + adjective."
9. I suggest... some more mathematical puzzles.

C. doing is the correct answer. The structure is "suggest + gerund."
10. We regret... you that we cannot approve your suggestion.

C. informing is the correct answer. The structure is "regret + gerund."