The key rule is: a speaker tells the truth when addressing someone older, and lies when addressing someone younger.

1. Assume statement 1 is true (Ethan to Lucas: "You're the youngest"). Then Lucas is youngest. But statement 3 says Lucas's age is the average of Sophia's and Ethan's ages — which would place Lucas between them, not youngest. Contradiction. So statement 1 must be false.

2. Since statement 1 is false, we ask: is this consistent with the rule? Ethan is speaking to Lucas. If Ethan > Lucas, then Ethan speaks to someone younger and must lie — consistent with statement 1 being false.

3. Let Sophia's age be \(s\). Assuming statements 2 and 3 are true (Sophia speaks to Ethan and Lucas, who are both older — so Sophia tells the truth):
• From statement 2: Ethan's age = \(1.6s\).
• From statement 3: Lucas's age = \(\frac{s + 1.6s}{2} = 1.3s\).
• Age order: Sophia \((s)\) < Lucas \((1.3s)\) < Ethan \((1.6s)\), confirming Sophia is youngest and speaks truthfully to both. ✓

4. For all ages to be distinct positive integers, \(1.6s = \frac{8s}{5}\) and \(1.3s = \frac{13s}{10}\) must both be integers, so \(s\) must be a positive multiple of \(10\).

5. Statement 4: Lucas \((1.3s)\) tells Sophia \((s)\): "I'm at least 5 years older." Since Lucas > Sophia, Lucas speaks to someone younger and must lie. The actual gap is \(0.3s\). For this to be a lie, we need \(0.3s < 5\), i.e. \(s < 16.\overline{6}\). The only positive multiple of \(10\) satisfying this is \(s = 10\).

Verification with \(s = 10\): Sophia = 10, Ethan = 16, Lucas = 13.
• Statement 1 (Ethan to Lucas): "Lucas is youngest" — false (Sophia is youngest). Ethan (16) > Lucas (13), so Ethan lies. ✓
• Statement 2 (Sophia to Ethan): "Your age is 60% greater" — true (\(10 \times 1.6 = 16\)). Sophia (10) < Ethan (16), so Sophia tells truth. ✓
• Statement 3 (Sophia to Lucas): "Your age is the average" — true (\((10 + 16)/2 = 13\)). Sophia (10) < Lucas (13), so Sophia tells truth. ✓
• Statement 4 (Lucas to Sophia): "At least 5 years older" — false (gap is \(13 - 10 = 3\)). Lucas (13) > Sophia (10), so Lucas lies. ✓

Therefore, Sophia is \(10\) years old.