Concept:
- When someone reports a suggestion using 'proposed/suggested that... should', and the suggestion includes the speaker along with others, the direct speech version uses the pattern "Let us + base verb".
- 'Let me' is used only when the suggestion is for the speaker alone, 'let them' is used when the speaker is excluded from the group.
Step 1: Read the indirect sentence carefully
"He proposed that they should wait for the award" is a group suggestion, and since he is proposing it together with the group he is part of, the direct version should include the speaker too.
Step 2: Recall the fixed structure for such suggestions
Proposals or suggestions of this type convert to "Let us + verb" in direct speech when the speaker is included in the action.
Step 3: Eliminate the options that do not fit this structure
'Let me wait' excludes everyone but the speaker, 'let them all wait' and 'let each one wait' exclude the speaker himself, none of these match a proposal that includes the speaker.
Step 4: Pick the correct direct speech
"Let us wait for the award" naturally reflects a proposal that includes the speaker along with the group.
Final Answer: He said, "Let us wait for the award."