3. Find all infinite positive real periodic sequences $X_{1}$, $X_{2}, \cdots$ that satisfy: for $n \in \mathbf{Z}_{+}$,
$$
X_{n+2}=\frac{1}{2}\left(\frac{1}{X_{n+1}}+X_{n}\right) \text {. }
$$
3. The required sequence is $X_{n}=1\left(n \in \mathbf{Z}_{+}\right)$.

If the smallest positive period of the sequence $T \geqslant 2$, denote a segment of length $T$ as $X_{1}, \cdots, X_{T}(T \geqslant 2)$.

By the condition $2 X_{i+2}=X_{i}+\frac{1}{X_{i+1}}$, summing for $i=1,2, \cdots, T$ we get
$$
X_{1}+X_{2}+\cdots+X_{T}=\frac{1}{X_{1}}+\frac{1}{X_{2}}+\cdots+\frac{1}{X_{T}} \text {. }
$$

Assume without loss of generality that $X_{1}=\max \left\{X_{1}, X_{2}, \cdots, X_{T}\right\}$. Then
$$
\begin{array}{l} 
X_{3}-\frac{1}{X_{2}}=X_{1}-X_{3} \geqslant 0 \\
\Rightarrow X_{2} X_{3} \geqslant 1 \\
\Rightarrow X_{3} X_{4}=\frac{1}{2} X_{3}\left(\frac{1}{X_{3}}+X_{2}\right) \\
=\frac{1}{2}+\frac{1}{2} X_{2} X_{3} \geqslant 1 .
\end{array}
$$

Thus, $X_{i} X_{i+1} \geqslant 1(i=1,2, \cdots, T)$, which means
$$
X_{i} \geqslant \frac{1}{X_{i+1}}(i=1,2, \cdots, T) \text {. }
$$

Summing for $i=1,2, \cdots, T$, we have
$$
X_{1}+X_{2}+\cdots+X_{T} \geqslant \frac{1}{X_{1}}+\frac{1}{X_{2}}+\cdots+\frac{1}{X_{T}} \text {. }
$$

Combining with equation (1), we get $X_{1}=X_{2}=\cdots=X_{T}=1$.
This contradicts $T \geqslant 2$.
Therefore, $T=1$.
Hence, $X_{n}=1\left(n \in \mathbf{Z}_{+}\right)$.