Six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries...

Understanding the Question

Let's break down what we're asked to determine:

• 6 countries sent a total of 75 representatives
• No two countries sent the same number
• Country A sent the second greatest number
• Question: Did Country A send at least 10 representatives?

This is a yes/no question. We need to determine definitively either:

• YES: Country A sent \(\geq 10\) representatives, or
• NO: Country A sent \(< 10\) representatives

Let's label the countries by their ranking:

• Country 1: Most representatives
• Country A: Second most representatives
• Countries 3, 4, 5, 6: The remaining four countries in descending order

So we have: \(\mathrm{Country\ 1} > \mathrm{Country\ A} > \mathrm{Country\ 3} > \mathrm{Country\ 4} > \mathrm{Country\ 5} > \mathrm{Country\ 6}\)

Analyzing Statement 1

What Statement 1 Tells Us

One of the six countries sent 41 representatives to the congress.

Testing Different Scenarios

Let's test which country could have sent 41 representatives:

Case 1: Country 1 sent 41

• Country A \(< 41\) (since Country A ranks second)
• The other 5 countries together sent: \(75 - 41 = 34\) representatives
• Since Country A is the largest among these 5 countries, Country A could theoretically be anywhere from 7 to 33
• Could Country A = 9? Yes! Example: 41, 9, 8, 7, 6, 3 (total = 75)
• Could Country A = 10? Yes! Example: 41, 10, 9, 8, 5, 2 (total = 75)
• Since Country A can be both \(< 10\) and \(\geq 10\), we get different answers (NO and YES)

Case 2: Country A sent 41

• Country 1 \(> 41\), so Country 1 \(\geq 42\)
• But \(42 + 41 = 83\), which already exceeds our total of 75
• This is impossible!

Case 3: One of Countries 3-6 sent 41

• If any lower-ranked country sent 41, then Country A \(> 41\)
• This means Country A \(\geq 42\), which is definitely \(\geq 10\)
• This would give us answer YES

Conclusion for Statement 1

Since different scenarios give different answers (Case 1 allows both YES and NO, while Case 3 gives only YES), Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

What Statement 2 Provides

Country A sent fewer than 12 representatives, so Country A \(\leq 11\).

Testing the Critical Values

Since Country A \(\leq 11\), let's test if Country A could be 9, 10, or 11:

Can Country A = 9? (This would give answer NO)

• If Country A = 9, then Country 1 \(\geq 10\)
• Example distribution: 40, 9, 8, 7, 6, 5 (total = 75) ✓
• All different ✓, Country A is second ✓
• Answer: NO

Can Country A = 10? (This would give answer YES)

• If Country A = 10, then Country 1 \(\geq 11\)
• Example distribution: 35, 10, 9, 8, 7, 6 (total = 75) ✓
• All different ✓, Country A is second ✓
• Answer: YES

Conclusion for Statement 2

Since Country A can be both \(< 10\) (giving answer NO) and \(\geq 10\) (giving answer YES), Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Both Statements

Combined Information

From both statements together:

• One country sent 41 representatives
• Country A \(< 12\) (so Country A \(\leq 11\))

Key Insight

Given Statement 2, Country A can only be 9, 10, or 11. The question becomes: does knowing "one country sent 41" narrow this down to a single answer?

Testing Combined Scenarios

Since Country A \(\leq 11\), and one country sent 41, that country must be Country 1 (the highest-ranking country).

Let's construct examples with Country 1 = 41:

Example 1 - Country A = 11:
Distribution: 41, 11, 10, 7, 4, 2

• Check: All different ✓, Total = 75 ✓, Country A is second ✓
• Country A = 11 \(\geq 10\)
• Answer: YES

Example 2 - Country A = 10:
Distribution: 41, 10, 9, 8, 5, 2

• Check: All different ✓, Total = 75 ✓, Country A is second ✓
• Country A = 10 \(\geq 10\)
• Answer: YES

Example 3 - Country A = 9:
Distribution: 41, 9, 8, 7, 6, 4

• Check: All different ✓, Total = 75 ✓, Country A is second ✓
• Country A = 9 \(< 10\)
• Answer: NO

Why the Statements Together Aren't Sufficient

Even with both statements, we've proven that Country A can still be either \(< 10\) or \(\geq 10\), giving different answers to our yes/no question. The constraint that one country sent 41 doesn't eliminate any of our possibilities for Country A (9, 10, or 11).

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

The statements together are not sufficient because we can construct valid scenarios where:

• Country A sent 9 representatives (answer NO)
• Country A sent 10 or 11 representatives (answer YES)

Since we cannot determine a definitive YES or NO answer even with both statements, the answer is E.

Answer Choice E: "The statements together are not sufficient."