The polynomial of seven variables
$$
Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)
$$
is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:
$$
Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.
$$
Find all possible values of $P_1(1,1,\ldots,1)$.

[i](A. Yuran)[/i]
1. **Express the polynomial \( Q(x_1, x_2, \ldots, x_7) \) in expanded form:**
   \[
   Q(x_1, x_2, \ldots, x_7) = (x_1 + x_2 + \ldots + x_7)^2 + 2(x_1^2 + x_2^2 + \ldots + x_7^2)
   \]
   Expanding \((x_1 + x_2 + \ldots + x_7)^2\):
   \[
   (x_1 + x_2 + \ldots + x_7)^2 = x_1^2 + x_2^2 + \ldots + x_7^2 + 2 \sum_{1 \leq i < j \leq 7} x_i x_j
   \]
   Therefore,
   \[
   Q(x_1, x_2, \ldots, x_7) = x_1^2 + x_2^2 + \ldots + x_7^2 + 2 \sum_{1 \leq i < j \leq 7} x_i x_j + 2(x_1^2 + x_2^2 + \ldots + x_7^2)
   \]
   Simplifying,
   \[
   Q(x_1, x_2, \ldots, x_7) = 3(x_1^2 + x_2^2 + \ldots + x_7^2) + 2 \sum_{1 \leq i < j \leq 7} x_i x_j
   \]

2. **Express \( Q(x_1, x_2, \ldots, x_7) \) as the sum of squares of polynomials:**
   \[
   Q(x_1, x_2, \ldots, x_7) = P_1(x_1, x_2, \ldots, x_7)^2 + P_2(x_1, x_2, \ldots, x_7)^2 + \ldots + P_7(x_1, x_2, \ldots, x_7)^2
   \]

3. **Analyze the coefficients of \( x_i^2 \) and \( x_i x_j \):**
   - Each \( x_i^2 \) has a coefficient of 3 in \( Q(x_1, x_2, \ldots, x_7) \).
   - Each \( x_i x_j \) has a coefficient of 2 in \( Q(x_1, x_2, \ldots, x_7) \).

4. **Determine the structure of \( P_i \):**
   - Since each \( x_i^2 \) has a coefficient of 3, each \( x_i \) must appear in exactly 3 of the \( P_i \)'s.
   - For each pair \( x_i x_j \), it must appear in exactly one \( P_i \).

5. **Let \( a_i \) be the number of variables in \( P_i \):**
   - Since each \( x_i \) appears in exactly 3 of the \( P_i \)'s, we have:
     \[
     \sum_{i=1}^7 a_i = 3 \times 7 = 21
     \]
   - The number of pairs \( (x_i, x_j) \) in \( P_i^2 \) is \( \frac{a_i(a_i - 1)}{2} \). Summing over all \( P_i \):
     \[
     \sum_{i=1}^7 \frac{a_i(a_i - 1)}{2} = \frac{7 \times 6}{2} = 21
     \]

6. **Use the Cauchy-Schwarz inequality:**
   \[
   \sum_{i=1}^7 a_i^2 \geq \frac{(\sum_{i=1}^7 a_i)^2}{7} = \frac{21^2}{7} = 63
   \]
   Since \( \sum_{i=1}^7 a_i^2 = 63 \), equality holds, implying \( a_i = 3 \) for all \( i \).

7. **Evaluate \( P_i(1, 1, \ldots, 1) \):**
   - Since each \( P_i \) contains exactly 3 variables, substituting \( x_1 = x_2 = \ldots = x_7 = 1 \):
     \[
     P_i(1, 1, \ldots, 1) = 3
     \]

Conclusion:
\[
\boxed{3}
\]