Problem 1829

To find all polynomials $P(x)$ with integer coefficients that satisfy the given condition, we analyze the condition: if $P(s)$ and $P(t)$ are integers for real numbers $s$ and $t$, then $P(st)$ must also be an integer.

### Step 1: Analyze the Degree of Polynomial

Assume $P(x) = a_d x^d + a_{d-1} x^{d-1} + \cdots + a_1 x + a_0$ where $a_i$ are integer coefficients.

The condition implies that for any real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are integers, then $P(st)$ is also an integer. Consider the simplest cases:

- Constant Polynomial: If $P(x) = c$ (a constant polynomial), then clearly $P(s) = P(t) = P(st) = c$, which is an integer. Thus, constant polynomials satisfy the condition.
- Linear Polynomial: Consider $P(x) = ax + b$.
- If $P(s) = as + b$ and $P(t) = at + b$ are integers, $P(st) = ast + b$ must also be an integer. This imposes no new constraints as $a, b$ are integers.

### Step 2: Consider Higher Degree Polynomials

- If $P(x) = a x^d + \cdots + c$ with $d \geq 1$, analyze whether such a polynomial can satisfy the condition:
- Let $P(s) = a s^d + \ldots + c$ and $P(t) = a t^d + \ldots + c$.
- The multiplication condition $P(st)$ being an integer suggests that formulating such a polynomial while maintaining integer values involves specific form.

A key insight here is that the presence of cross-terms in the polynomial at higher degrees might violate integer preservation without specific structures.

### Step 3: Structure Imposition

If $P(x) = x^d + c$ or $P(x) = -x^d + c$, then:
- $P(s) = s^d + c$ and $P(t) = t^d + c$ are integers assuming they yield integers separately.
- Consequently, if both $P(s)$ and $P(t)$ are integers, then:
$$P(st) = (st)^d + c = s^d t^d + c$$
remains an integer because $s^d$ and $t^d$ are integers.

This structure ensures that $P(x) = \pm x^d + c$, thereby fulfilling the requirements.

### Conclusion

Thus, the form of the polynomial that satisfies the condition is:
$$P(x) = \pm x^d + c$$
where $c$ is an integer and $d$ is a positive integer.

Hence, the final answer is:
$$\boxed{P(x)=\pm x^d+c \text{, where } c \text { is an integer and } d \text{ is a positive integer.}}$$