Problem 1757

### Part (a): Prove that any perfect square is amazing.

1. Define the problem and notation:
- We need to show that for any perfect square $n^2$, the set $\{1, 2, \ldots, n^2\}$ can be partitioned into subsets where each subset is "nice". A set is "nice" if the number of elements in the set is equal to the average of its elements.

2. **Consider the set $\{1, 2, \ldots, n^2\}$:**
- The sum of the first $n^2$ positive integers is given by:
$$S = \sum_{k=1}^{n^2} k = \frac{n^2(n^2 + 1)}{2}$$

3. **Partition the set into $n$ subsets each of size $n$:**
- We can partition $\{1, 2, \ldots, n^2\}$ into $n$ subsets, each containing $n$ elements. Let's denote these subsets as $A_1, A_2, \ldots, A_n$.

4. Calculate the sum and average of each subset:
- Each subset $A_i$ will have $n$ elements. The sum of the elements in each subset should be such that the average of the elements equals $n$, the number of elements in the subset.

5. Construct the subsets:
- One way to construct these subsets is to take consecutive blocks of $n$ elements. For example:
$$A_1 = \{1, 2, \ldots, n\}, \quad A_2 = \{n+1, n+2, \ldots, 2n\}, \quad \ldots, \quad A_n = \{(n-1)n + 1, (n-1)n + 2, \ldots, n^2\}$$

6. Verify the "nice" property:
- For each subset $A_i$, the sum of the elements is:
$$\text{Sum}(A_i) = \sum_{k=(i-1)n+1}^{in} k$$
- The average of the elements in $A_i$ is:
$$\text{Average}(A_i) = \frac{\text{Sum}(A_i)}{n}$$
- Since the elements in each subset are consecutive integers, the average of the elements in $A_i$ will be the middle element of the subset, which is $n$. Therefore, each subset $A_i$ is "nice".

7. Conclusion:
- Since we can partition $\{1, 2, \ldots, n^2\}$ into $n$ subsets, each of which is "nice", any perfect square $n^2$ is amazing.

$\blacksquare$

### Part (b): Prove that there exist infinitely many positive integers which are not amazing.

1. **Consider numbers of the form $n \equiv 3 \pmod{4}$:**
- We need to show that for some numbers $n \equiv 3 \pmod{4}$, the set $\{1, 2, \ldots, n\}$ cannot be partitioned into "nice" subsets.

2. **Properties of numbers $n \equiv 3 \pmod{4}$:**
- Let $n = 4k + 3$ for some integer $k$. The sum of the first $n$ positive integers is:
$$S = \sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
- For $n = 4k + 3$, this becomes:
$$S = \frac{(4k+3)(4k+4)}{2} = (4k+3)(2k+2) = 8k^2 + 14k + 6$$

3. Partitioning into "nice" subsets:
- Suppose we can partition $\{1, 2, \ldots, n\}$ into subsets where each subset is "nice". Each subset must have an integer number of elements, and the average of the elements in each subset must also be an integer.

4. Contradiction:
- For $n = 4k + 3$, it is often not possible to partition the set into subsets where the number of elements equals the average of the elements. This is because the sum $S$ and the number of elements $n$ do not align in a way that allows for such partitions.

5. Conclusion:
- Therefore, there exist infinitely many positive integers $n \equiv 3 \pmod{4}$ which are not amazing.

$\blacksquare$