Analyzing Exam Score Logic

This solution analyzes three statements about student exam scores to determine which logical implication necessarily holds true.

Understanding the Statements

Let the student scores be ordered in descending order: $S_1 \ge S_2 \ge S_3 \ge S_4 \ge \dots$. The maximum possible score is 150.

• (S1) The highest score is 100. This means $S_1 = 100$.
• (S2) The fourth highest score is 76. This means $S_4 = 76$.
• (S3) There are at least four students whose scores are within 25 of each other. This means there exist four scores $s_a, s_b, s_c, s_d$ such that $\max(s_a, s_b, s_c, s_d) - \min(s_a, s_b, s_c, s_d) \le 25$.

Evaluating Option A: (S1) and (S2) imply (S3)

Let's assume statements (S1) and (S2) are true.

• From (S1), the highest score is $S_1 = 100$.
• From (S2), the fourth highest score is $S_4 = 76$.
• This gives us the four highest scores: $100, S_2, S_3, 76$. We know that $100 \ge S_2 \ge S_3 \ge 76$.
• Consider these four scores: $100, S_2, S_3,$ and $76$.
• The maximum score among these four is 100.
• The minimum score among these four is 76.
• The difference between the maximum and minimum of these four scores is $100 - 76 = 24$.
• Since $24 \le 25$, these four scores ($100, S_2, S_3,$ and $76$) are within 25 of each other.
• This directly satisfies the condition stated in (S3).

Therefore, statements (S1) and (S2) together necessarily imply statement (S3).

Evaluating Other Options

Let's briefly check why other options are not necessarily correct.

• Option B ((S1) and (S3) imply (S2)): Consider scores {100, 90, 85, 80}. S1 is true (100). S3 is true ({100, 90, 85, 80} range is $100-80=20 \le 25$). But S2 is false (fourth score is 80, not 76). So, this is not necessarily correct.
• Option C ((S2) and (S3) imply (S1)): Consider scores {110, 80, 79, 77, 76}. S2 is true (76). S3 is true ({80, 79, 77, 76} range is $80-76=4 \le 25$). But S1 is false (highest score is 110, not 100). So, this is not necessarily correct.
• Option D ((S1) implies (S3)): Consider scores {100, 70, 60, 50}. S1 is true (100). S3 is false (no four scores are within 25; $100-50=50$). So, this is not necessarily correct.

Conclusion

Based on the analysis, only the combination of statements (S1) and (S2) guarantees that statement (S3) is true.