Determine the largest positive integer $n$ such that the following statement is true:

There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.
1. **Claim and Construction for \( n = 3 \)**:
   - We claim that \( n = 3 \) is the largest positive integer such that the given conditions hold.
   - Consider the polynomials \( P_1(x) = 1 \), \( P_2(x) = 2 \), and \( P_3(x) = -3 \).
   - The sum of any two of these polynomials is:
     \[
     P_1(x) + P_2(x) = 1 + 2 = 3, \quad P_1(x) + P_3(x) = 1 - 3 = -2, \quad P_2(x) + P_3(x) = 2 - 3 = -1
     \]
     None of these sums have real roots since they are constant non-zero polynomials.
   - The sum of all three polynomials is:
     \[
     P_1(x) + P_2(x) + P_3(x) = 1 + 2 - 3 = 0
     \]
     This sum has a real root at \( x = 0 \).

2. **Proof that \( n \geq 4 \) Fails**:
   - Assume for contradiction that \( n \geq 4 \) and there exist polynomials \( P_1(x), P_2(x), P_3(x), P_4(x) \) satisfying the conditions.
   - Define \( p_i = P_i(0) \) for \( i = 1, 2, 3, 4 \).
   - Without loss of generality, assume \( |p_1| \) is the largest among \( |p_1|, |p_2|, |p_3|, |p_4| \).
   - Since \( P_1(x) + P_2(x) \), \( P_1(x) + P_3(x) \), and \( P_1(x) + P_4(x) \) have no real roots, their constant terms \( p_1 + p_2 \), \( p_1 + p_3 \), and \( p_1 + p_4 \) must all have the same sign. Assume without loss of generality that this sign is positive.
   - Suppose \( p_2 + p_3 \) is positive. Then:
     \[
     P_1(x) + P_2(x) > 0, \quad P_2(x) + P_3(x) > 0, \quad P_1(x) + P_3(x) > 0 \quad \text{for all } x
     \]
     Summing these inequalities:
     \[
     (P_1(x) + P_2(x)) + (P_2(x) + P_3(x)) + (P_1(x) + P_3(x)) > 0 + 0 + 0
     \]
     Simplifying:
     \[
     2(P_1(x) + P_2(x) + P_3(x)) > 0 \implies P_1(x) + P_2(x) + P_3(x) > 0 \quad \text{for all } x
     \]
     This implies \( P_1(x) + P_2(x) + P_3(x) \) has no real roots, contradicting the given condition.
   - Therefore, \( p_2 + p_3 \), \( p_3 + p_4 \), and \( p_2 + p_4 \) must all be negative. Using a similar argument:
     \[
     P_2(x) + P_3(x) < 0, \quad P_3(x) + P_4(x) < 0, \quad P_2(x) + P_4(x) < 0 \quad \text{for all } x
     \]
     Summing these inequalities:
     \[
     (P_2(x) + P_3(x)) + (P_3(x) + P_4(x)) + (P_2(x) + P_4(x)) < 0 + 0 + 0
     \]
     Simplifying:
     \[
     2(P_2(x) + P_3(x) + P_4(x)) < 0 \implies P_2(x) + P_3(x) + P_4(x) < 0 \quad \text{for all } x
     \]
     This implies \( P_2(x) + P_3(x) + P_4(x) \) has no real roots, again contradicting the given condition.

Thus, \( n = 3 \) is the largest positive integer for which the given conditions hold.

The final answer is \( \boxed{ n = 3 } \)