1.Peace activist Baroness Bertha von Suttner encouraged Alfred Nobel ... a prize for peace. (establish / to establish / establishing / to be established)
2...Fred couldn’t make … that he had caught the huge fish himself. (I believed/ me believe / me believed / me to believe)
3 They wouldn't let ... the country without the entry visa, would they? (him to enter / he enters / him enter / him entering)
4. Nick was very close to the phone-box and Mary noticed him … the number, and then she heard him ... something to passer-by. (to dial, to say / dialing, to say / to dial, say / dial, say)
5.The police officer made Neil get out of the car and demanded … his driver's licence.(see / seeing / to see / to be seen)
6.In the time of Peter the Great the noblemen were not allowed … beards. (have / to have / having / having had)

1. Peace activist Baroness Bertha von Suttner encouraged Alfred Nobel to establish a prize for peace.
Explanation: The correct verb form here is "to establish" because it is used after the verb "encouraged". When we use the verb "encourage", we follow it with the infinitive form of the verb.

2. I believed Fred couldn’t make me believe that he had caught the huge fish himself.
Explanation: The correct form here is "I believed" because it is the past tense form of the verb "believe". The sentence talks about the speaker's belief in the past.

3. They wouldn't let him enter the country without the entry visa, would they?
Explanation: The correct form here is "him enter" because it is the correct structure when using the verb "let". After "let", we follow it with the object pronoun and the bare infinitive form of the verb.

4. Nick was very close to the phone-box and Mary noticed him dialing the number, and then she heard him say something to a passer-by.
Explanation: The correct forms here are "dialing" and "say" because they are the present participle form and the base form of the verb "hear" respectively. When using the verb "hear", we follow it with the present participle form of the verb to indicate an ongoing action.

5. The police officer made Neil get out of the car and demanded to see his driver's license.
Explanation: The correct form here is "to see" because it follows the verb "demanded". After "demand", we use the infinitive form of the verb.

6. In the time of Peter the Great, the noblemen were not allowed to have beards.
Explanation: The correct form here is "to have" because it follows the verb "allowed". After "allow", we use the infinitive form of the verb.