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The problem is asking to find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the given functional equation:

\[
f(x + f(y)) = f(x + xy) + y f(1 - x) \quad \forall x, y \in \mathbb{R}
\]

### Approach

1. **Initial Observation**: This functional equation is defined for all real numbers \(x\) and \(y\), meaning we can substitute specific values for \(x\) and \(y\) to simplify the equation and gain insights into the behavior of the function \(f\).

2. **Test with \( y = 0 \)**:
   \[
   f(x + f(0)) = f(x)
   \]
   This suggests that \(f(0)\) could be an important value to determine. Let's keep this observation in mind.

3. **Test with \( x = 1 \)**:
   \[
   f(1 + f(y)) = f(1 + y) + y f(0)
   \]
   This gives a relation between \(f(1 + f(y))\) and \(f(1 + y)\), which might help us establish further properties of the function \(f\).

4. **Test with \( y = 1 \)**:
   \[
   f(x + f(1)) = f(x + x) + f(1 - x)
   \]
   This could help determine \(f(1)\) if we analyze the behavior for specific values of \(x\).

By continuing substitutions for specific \(x\) and \(y\), we aim to simplify the equation further, allowing us to discover the exact form of the function \(f(x)\).

Would you like to continue with this detailed step-by-step exploration, or would you prefer a more direct approach to find the solution?

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Here are some questions to consider:
1. What happens when we assume \( f(x) = x \) or other linear forms?
2. Can we find the value of \( f(0) \) directly from the equation?
3. How does the equation behave for negative values of \( x \) or \( y \)?
4. What specific properties of functional equations are useful for solving this problem?
5. Can we derive symmetry or periodicity from the equation?

**Tip**: In functional equations, substituting special values like \(x = 0\) or \(y = 0\) often simplifies the equation and reveals important properties of the function.

Find all functions f: ℝ → ℝ such that f(x + f(y)) = f(x + xy) + y f(1 - x) for all x, y ∈ ℝ.

Learn how to find all functions f: ℝ → ℝ that satisfy the equation f(x + f(y)) = f(x + xy) + y f(1 - x) for all x, y ∈ ℝ. This problem explores concepts in functional equations, algebra, and real analysis, using techniques like substitution to find the solution.

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