Understanding the Weight Function Behavior

The problem describes a weight function, $W(a)$, which depends on age, $a$. The function exhibits the following properties:

• It monotonically increases for ages up to 50 years ($a \le 50$).
• It monotonically decreases for ages after 50 years ($a \ge 50$).

This means the weight typically peaks around age 50.

Analyzing Brothers' Ages for Equal Weight

Let the ages of the two brothers be $a_1$ and $a_2$, with the condition $a_1 < a_2$. They achieve the same weight, so $W(a_1) = W(a_2)$. We evaluate the possibilities:

Case 1: Both Ages Below 50

If $a_1 < a_2 < 50$, the weight function is strictly increasing. Thus, $W(a_1) < W(a_2)$. They cannot have the same weight.

Case 2: Both Ages Above 50

If $50 < a_1 < a_2$, the weight function is strictly decreasing. Thus, $W(a_1) > W(a_2)$. They cannot have the same weight.

Case 3: One Age is Exactly 50

• If $a_1 = 50$ (so $a_2 > 50$): $W(a_1)$ is the maximum weight. Since the function decreases after 50, $W(a_2) < W(50) = W(a_1)$. They cannot have the same weight.
• If $a_2 = 50$ (so $a_1 < 50$): $W(a_2)$ is the maximum weight. Since the function increases until 50, $W(a_1) < W(50) = W(a_2)$. They cannot have the same weight.

Case 4: One Age Below 50, One Age Above 50

The only scenario where $W(a_1) = W(a_2)$ can occur with $a_1 < a_2$ is if one age is on the increasing part of the curve and the other is on the decreasing part. This requires $a_1 < 50$ and $a_2 > 50$. This situation satisfies the condition $a_1 < 50 < a_2$.

Final Deduction

Given the properties of the weight function and the condition $a_1 < a_2$, the only possibility for the brothers to attain the same weight is when one brother's age is less than 50 years and the other's age is greater than 50 years.

Therefore, the correct statement is $a_1 < 50 < a_2$.