To determine if the given triplets of polynomials are *smart*, we need to check if there exist polynomials $P, Q, R \in \mathbb{R}[x,y,z]$ such that the polynomial identity $u^{2019}P + v^{2019}Q + w^{2019}R = 2019$ holds.

### Part (a)
Given the triplet:
$$u = x + 2y + 3, \quad v = y + z + 2, \quad w = x + y + z$$

1. Formulate the polynomial identity:
We need to find polynomials $P, Q, R$ such that:
$$(x + 2y + 3)^{2019}P + (y + z + 2)^{2019}Q + (x + y + z)^{2019}R = 2019$$

2. Analyze the degrees:
Each term $u^{2019}, v^{2019}, w^{2019}$ is a polynomial of degree 2019. For the left-hand side to equal the constant 2019, the polynomials $P, Q, R$ must be chosen such that the combination of terms results in a constant.

3. Check for possible solutions:
Since $u, v, w$ are linear polynomials, their 2019th powers will be very high-degree polynomials. It is highly unlikely that a combination of such high-degree polynomials can result in a constant term without specific constraints on $P, Q, R$.

4. Conclusion:
It is not possible to find such polynomials $P, Q, R$ that satisfy the given identity for the triplet $u = x + 2y + 3, v = y + z + 2, w = x + y + z$.

### Part (b)
Given the triplet:
$$u = x + 2y + 3, \quad v = y + z + 2, \quad w = x + y - z$$

1. Formulate the polynomial identity:
We need to find polynomials $P, Q, R$ such that:
$$(x + 2y + 3)^{2019}P + (y + z + 2)^{2019}Q + (x + y - z)^{2019}R = 2019$$

2. **Rewrite $u$ in terms of $v$ and $w$:**
Notice that:
$$u = (v - z + 1) + (w + z) = v + w + 1$$

3. Substitute and simplify:
We need to find $P \in \mathbb{R}[v, w]$ such that:
$$(v + w + 1)^{2019}P - 1$$
can be written as a linear combination of $v^{2019}Q$ and $w^{2019}R$.

4. **Choose a suitable $P$:**
Let $P(v, w) = T(v + w)$. We need $T$ such that:
$$(x + 1)T(x) - 1$$
is divisible by $x^{4038}$. This means:
$$(x + 1)T(x) = x^{4038}S(x) + 1$$

5. **Determine $S$:**
The only restriction on $S$ is that $S(-1) = -1$. We can choose $S(x) = -1$, giving:
$$T(x) = \frac{1 - x^{4038}}{1 + x}$$

6. Verify the solution:
With this choice of $T$, we have:
$$(v + w + 1)^{2019} \cdot \frac{1 - (v + w)^{4038}}{1 + (v + w)} - 1$$
which can be written as a linear combination of $v^{2019}Q$ and $w^{2019}R$.

7. Conclusion:
The triplet $u = x + 2y + 3, v = y + z + 2, w = x + y - z$ is *smart*.

The final answer is: