Medals Calculation and Distribution

Let's break down the problem step by step to figure out the total number of medal winners in the examination.

Total Medals Awarded

We are given that there are 5 subjects and 3 medals were awarded for each subject. To find the total number of medals awarded, we multiply the number of subjects by the number of medals per subject:

\( \text{Total medals} = \text{Number of subjects} \times \text{Medals per subject} \)

\( \text{Total medals} = 5 \times 3 = 15 \)

So, a total of 15 medals were awarded.

Medals Won by Specific Candidates

The problem states that three candidates won exactly four medals each. Let's calculate the total number of medals these three candidates won:

\( \text{Medals won by three candidates} = \text{Number of candidates} \times \text{Medals per candidate} \)

\( \text{Medals won by three candidates} = 3 \times 4 = 12 \)

These three candidates account for 12 of the 15 total medals.

Remaining Medals

Now, we need to find out how many medals are left to be distributed among any other potential winners. We subtract the medals won by the first three candidates from the total medals:

\( \text{Remaining medals} = \text{Total medals} - \text{Medals won by three candidates} \)

\( \text{Remaining medals} = 15 - 12 = 3 \)

There are 3 medals remaining.

Constraint on Medal Winners

A crucial piece of information is that "no candidate won just one medal". This means any candidate who won a medal must have won 2 or more medals.

Distribution of Remaining Medals

The remaining 3 medals must be won by other candidates, besides the initial three who won 4 medals each. Let's consider how these 3 medals can be distributed among additional winners, keeping in mind that each additional winner must receive at least 2 medals:

• If one additional candidate wins the remaining medals, they would win 3 medals. Winning 3 medals satisfies the condition of not winning just one medal (since \(3 > 1\)). This scenario is possible.
• If two or more additional candidates were to win the remaining 3 medals, each would need to win at least 2 medals. For example, if two candidates won, the minimum number of medals they could win together is \(2 + 2 = 4\). Since there are only 3 medals remaining, it is impossible for two or more additional candidates to share these medals if each must receive at least 2.

Therefore, the only way to distribute the remaining 3 medals under the given conditions is for one additional candidate to win all 3 of them.

Total Number of Medal Winners

We have the initial three candidates who won exactly 4 medals each, and we have determined that there must be exactly one additional candidate who won the remaining 3 medals. So, the total number of medal winners is:

\( \text{Total medal winners} = \text{Candidates with 4 medals} + \text{Additional candidates with medals} \)

\( \text{Total medal winners} = 3 + 1 = 4 \)

Based on the problem constraints, the total number of medal winners must be exactly 4.

This conclusion matches option 1.