Question

Six square states of equal area are arranged in two columns running north to south in a country. State 1, State 3, and State 5 lie on the western side, from north to south, and State 2, State 4, and State 6 lie on the eastern side, from north to south, so the states directly across the street from each other are State 1 and State 2, State 3 and State 4, and State 5 and State 6. Two states share a boundary if they sit next to each other in the same column (north-south) or directly across from each other (east-west). Within these six states there are exactly eight institutions in total: four medical institutes, two management institutes, and two technical institutes, placed according to these rules:

1. No institution is located in more than one state.
2. No state contains more than one management institute, and no state contains more than one technical institute.
3. No state contains both a management institute and a technical institute.
4. Every management institute is located in a state that also contains at least one medical institute.
5. The two technical institutes are located in states that do not share a common boundary.
6. State 3 contains a technical institute, and State 6 contains a management institute.

If each of the six states contains at least one of the eight institutions, then which one of the following must be true?

• A. There is a management institute in State 1
• B. There is a medical institute in State 2
• C. There is a medical institute in State 3
• D. There is a medical institute in State 4

Hint:

Count how many medical and management institutes are left after fixing State 2, State 3 and State 6, then see how they must be shared among State 1, State 4 and State 5.

Answer 1

The Correct Option is D

Step 1: List what is already fixed and what is left to place.
State 2 and State 3 each hold one technical institute. State 6 holds one management institute and at least one medical institute.
Out of the 8 institutions, 4 medical, 2 management and 2 technical, we have placed 2 technical and 1 management so far, and 1 medical institute is already used up inside State 6.
So we still need to place 1 management institute and 3 medical institutes, and these must all go into State 1, State 4 and State 5, the three states with nothing fixed yet.

Step 2: Apply the new condition that every state must hold at least one institution.
State 1, State 4 and State 5 currently hold nothing, so each of them must receive at least one of the remaining 4 institutions, which are 1 management and 3 medical.
The leftover management institute must go into one of these three states, and rule 4 says it needs a medical institute in the same state. So that one state gets 2 institutions, management plus medical, using up 1 of the 3 remaining medical institutes.
That leaves exactly 2 medical institutes for the other 2 states, one each, since every state still needs at least one institution and there is nothing else left to give them.

Step 3: Check what this means for every one of State 1, State 4 and State 5.
Whichever one of the three states gets the management institute, it also gets a medical institute alongside it.
The remaining two states each get exactly one medical institute, since that is the only institution type left and each state needs at least one.
So no matter which of State 1, State 4 or State 5 ends up with the management institute, all three of them end up with at least one medical institute.

Step 4: Test each option.
Option A, management institute in State 1: only true in one of the three possible arrangements, not in all of them, so it is not guaranteed.
Option B, medical institute in State 2: false, State 2 already holds a technical institute, and giving it a medical institute too would leave one of State 1, State 4 or State 5 with nothing, breaking the new condition.
Option C, medical institute in State 3: false, for the same reason as State 2.
Option D, medical institute in State 4: true in every possible arrangement, since Step 3 shows all of State 1, State 4 and State 5 must get at least one medical institute.

Final Answer:
Only "there is a medical institute in State 4" holds in every valid arrangement. \[ \boxed{\text{Option D}} \]