Analyzing S and M's Statements

This problem requires us to determine the ages of two individuals, S and M, based on statements they make about their own and each other's ages. The crucial condition is that all statements made are false.

Breaking Down Each Statement

Let's denote the age of S as $A_S$ and the age of M as $A_M$. We will analyze each statement and its logical negation:

Statement 1 (by S): "I am certainly not over 40 year."

• The statement translates to: $A_S \le 40$.
• If this statement is wrong, the opposite must be true: $A_S > 40$. This means S's age must be at least 41.

Statement 2 (by M): "I am 38 years and you are at least 5 years older than me."

• This statement is a conjunction (AND) of two parts:
  • Part 2a: M's age is 38 ($A_M = 38$).
  • Part 2b: S is at least 5 years older than M ($A_S \ge A_M + 5$).
• The statement translates to: $A_M = 38 \land A_S \ge A_M + 5$.
• If this entire statement is wrong, its negation must be true. The negation of 'P AND Q' is 'NOT P OR NOT Q'.
• Therefore, the negation is: $A_M \ne 38 \lor A_S < A_M + 5$. This means either M is not 38, OR S is less than 5 years older than M (or both).

Statement 3 (by S): "You are at least 39 years."

• The statement translates to: $A_M \ge 39$.
• If this statement is wrong, the opposite must be true: $A_M < 39$. This means M's age must be 38 or less.

Deriving Core Conditions

From the negations of the statements, we have established the following necessary conditions for the ages $A_M$ and $A_S$:

• Condition 1 (from S's first statement): $A_S > 40$.
• Condition 2 (from S's second statement): $A_M < 39$.
• Condition 3 (from M's statement): $A_M \ne 38 \lor A_S < A_M + 5$.

Evaluating Possible Age Combinations

We need to find ages ($A_M, A_S$) that satisfy all three derived conditions. Let's explore possibilities based on Condition 2 ($A_M < 39$):

Case 1: M's age is exactly 38 ($A_M = 38$)

• This satisfies Condition 2 ($38 < 39$).
• Now, let's check Condition 3: $A_M \ne 38 \lor A_S < A_M + 5$.
  • Substituting $A_M = 38$, we get: $38 \ne 38 \lor A_S < 38 + 5$.
  • This simplifies to: False $\lor A_S < 43$.
  • For Condition 3 to be true, we must have $A_S < 43$.
• Combining with Condition 1 ($A_S > 40$), the possible integer ages for S are 41 or 42 when $A_M = 38$.
• Potential valid solutions found: $(A_M=38, A_S=41)$ and $(A_M=38, A_S=42)$.

Case 2: M's age is not 38 ($A_M \ne 38$)

• Since Condition 2 states $A_M < 39$, M's age must be 37 or less ($A_M \le 37$).
• In this scenario, the first part of Condition 3 ($A_M \ne 38$) is true.
• Therefore, Condition 3 ($A_M \ne 38 \lor A_S < A_M + 5$) is automatically satisfied, regardless of the value of $A_S < A_M + 5$.
• We only need to satisfy Condition 1 ($A_S > 40$) and the fact that $A_M \le 37$.
• Any pair $(A_M, A_S)$ where $A_M \le 37$ and $A_S \ge 41$ would be a valid solution.
• Examples include: $(A_M=37, A_S=41)$, $(A_M=36, A_S=41)$, etc. Note that $(A_M=36, A_S=41)$ corresponds to Option 2.

Conclusion on Determining Ages

We have found multiple possible age combinations that satisfy the condition that all original statements are false. For instance:

• $(A_M=38, A_S=41)$ is a valid solution satisfying all derived conditions.
• $(A_M=36, A_S=41)$ is also a valid solution satisfying all derived conditions.

Since there is more than one possible pair of ages for M and S that fits the given criteria, we cannot pinpoint a single, unique answer from the options provided. Therefore, the ages cannot be determined.