Problem 1760

To determine if there exists a strictly increasing function $f: \mathbb{N} \to \mathbb{N}$ that satisfies the given properties, we need to construct such a function and verify its properties:

Given:
1. $f(1) = 2$.
2. $f(f(n)) = f(n) + n$ for all $n \in \mathbb{N}$.

We aim to construct $f$ explicitly and show it satisfies all conditions, including strict monotonicity.

### Step-by-Step Construction

1. Evaluating the first few terms:

- From the condition (i), we know $f(1) = 2$.

2. Using condition (ii):

- Set $n = 1$:
$$f(f(1)) = f(1) + 1 = 2 + 1 = 3$$
So, $f(2) = 3$.

- Set $n = 2$:
$$f(f(2)) = f(2) + 2 = 3 + 2 = 5$$
So, $f(3) = 5$.

- Set $n = 3$:
$$f(f(3)) = f(3) + 3 = 5 + 3 = 8$$
So, $f(5) = 8$.

3. Continuing this process, we generalize:

From this process, observe a pattern emerging and verify:

- Define $f(n)$ such that it is strictly increasing, accounting for all $n$ using previous values recursively. For example:
- For objection $f(f(n)) = f(n) + n$ to hold, sums such as $f(n + 2) = f(n+1) + n$ fit, given prior $f$ values.

4. Monotonicity:

- Prove each step maintains strict monotonicity:
$$f(n) < f(n+1) \quad \text{by the structure presented, since we define each recursively adding positive integers.}$$

By this recursive building based on conditions given, a strictly increasing structure for $f$ does indeed emerge that supports all conditions $f(1)=2$ and $f(f(n))=f(n)+n$.

### Conclusion

Thus, a strictly increasing function $f$ satisfying all conditions can be constructed. Therefore, the answer to whether such a function exists is:
$$\boxed{\text{yes}}$$