Comprehension:

In a coaching class, some students register online, and some others register offline. No student registers both online and offline; hence the total registration number is the sum of online and offline registrations. The following facts and table pertain to these registration numbers for the five months - January to May of 2023. The table shows the minimum, maximum, median registration numbers of these five months, separately for online, offline and total number of registrations. The following additional facts are known.

1. In every month, both online and offline registration numbers were multiples of 10.

2. In January, the number of offline registrations was twice that of online registrations.

3. In April, the number of online registrations was twice that of offline registrations.

4. The number of online registrations in March was the same as the number of offline registrations in February.

5. The number of online registrations was the largest in May.

	Minimum	Maximum	Median
Online	40	100	80
Offline	30	80	50
Total	110	130	120

Which of the following statements can be true?

I. The number of offline registrations was the smallest in May.

II. The total number of registrations was the smallest in February.

To determine which statements can be true, let us analyze the given information step-by-step.

Let the online, offline, and total registrations for month m (where m represents January, February, March, April, or May) be denoted by $O_m$, $F_m$, and $T_m$ respectively. We are given that all registration numbers are multiples of 10 and that $T_m=O_m+F_m$ for every month.

From the given table, we can write the sorted values for Online, Offline, and Total registrations over the 5 months:
• Online: Min = 40, Median = 80, Max = 100.
• Offline: Min = 30, Median = 50, Max = 80.
• Total: Min = 110, Median = 120, Max = 130.

Now we apply the additional facts:
Fact 2: In January, offline is twice online:
$F_{\text{Jan}}=2O_{\text{Jan}}$
$T_{\text{Jan}}=O_{\text{Jan}}+F_{\text{Jan}}=3O_{\text{Jan}}$
Since $T_{\text{Jan}}$ must be a multiple of 10 between 110 and 130, the only multiple of 3 in this range is 120. Thus:
$T_{\text{Jan}}=120$, $O_{\text{Jan}}=40$, $F_{\text{Jan}}=80$.

Fact 3: In April, online is twice offline:
$O_{\text{Apr}}=2F_{\text{Apr}}$
$T_{\text{Apr}}=O_{\text{Apr}}+F_{\text{Apr}}=3F_{\text{Apr}}$
Similarly, $T_{\text{Apr}}$ must be 120. Thus:
$T_{\text{Apr}}=120$, $F_{\text{Apr}}=40$, $O_{\text{Apr}}=80$.

Fact 5: Online registrations were the largest in May.
Since the maximum online registration is 100, we must have:
$O_{\text{May}}=100$.

Fact 4: Online registrations in March equals offline registrations in February:
$O_{\text{Mar}}=F_{\text{Feb}}$.

Let us analyze the Online registrations. The five values are:
$O_{\text{Jan}}=40$, $O_{\text{Apr}}=80$, $O_{\text{May}}=100$, $O_{\text{Feb}}$, and $O_{\text{Mar}}$.
Since the median of the online registrations is 80, when sorted, the third value must be 80. Since we already have 40, 80, and 100, the remaining two values ($O_{\text{Feb}}$ and $O_{\text{Mar}}$) must be distributed such that:
One of them is in the range $[40,80]$ and the other is in the range $[80,100]$.

Now, let us evaluate the two statements:

Statement I: The number of offline registrations was the smallest in May.
This means $F_{\text{May}}=30$ (since 30 is the minimum offline registration value).
If $F_{\text{May}}=30$, then:
$T_{\text{May}}=O_{\text{May}}+F_{\text{May}}=100+30=130$ (which is consistent with the maximum total).
Let us verify if a complete consistent configuration can be constructed.
Let $F_{\text{Feb}}=50$ (which satisfies the median condition for offline).
By Fact 4, $O_{\text{Mar}}=F_{\text{Feb}}=50$. This is in the range $[40,80]$, which is valid.
Let the other online registration $O_{\text{Feb}}=80$ (which is in the range $[80,100]$).
Then, $T_{\text{Feb}}=O_{\text{Feb}}+F_{\text{Feb}}=80+50=130$.
For March, let $F_{\text{Mar}}=60$, giving $T_{\text{Mar}}=O_{\text{Mar}}+F_{\text{Mar}}=50+60=110$.
Let us check the sorted lists:
• Online: 40 (Jan), 50 (Mar), 80 (Apr), 80 (Feb), 100 (May) → Min=40, Median=80, Max=100. (Valid)
• Offline: 30 (May), 40 (Apr), 50 (Feb), 60 (Mar), 80 (Jan) → Min=30, Median=50, Max=80. (Valid)
• Total: 110 (Mar), 120 (Jan), 120 (Apr), 130 (Feb), 130 (May) → Min=110, Median=120, Max=130. (Valid)
Since all conditions are satisfied, Statement I can be true.

Statement II: The total number of registrations was the smallest in February.
This means $T_{\text{Feb}}=110$.
Thus, we must have:
$O_{\text{Feb}}+F_{\text{Feb}}=110$
Since $F_{\text{Feb}}=O_{\text{Mar}}$, this becomes:
$O_{\text{Feb}}+O_{\text{Mar}}=110$
As established earlier, one of these online values must be in $[40,80]$ and the other in $[80,100]$.
Let the larger value be at least 80. Then the smaller value must be:
$\text{Smaller Value}=110-\text{Larger Value}\le 110-80=30$.
However, the minimum online registration is 40, meaning neither $O_{\text{Feb}}$ nor $O_{\text{Mar}}$ can be 30 or less.
Thus, it is mathematically impossible to have $O_{\text{Feb}}+O_{\text{Mar}}=110$.
Therefore, Statement II cannot be true.

In conclusion, only Statement I can be true.