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 (Bob to Charlie: "You're the youngest"). Then Charlie is youngest. But statement 3 says Charlie's age is the average of Alice's and Bob's ages — which would place Charlie 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? Bob is speaking to Charlie. If Bob > Charlie, then Bob speaks to someone younger and must lie — consistent with statement 1 being false.

3. Let Alice's age be \(s\). Assuming statements 2 and 3 are true (Alice speaks to Bob and Charlie, who are both older — so Alice tells the truth):
• From statement 2: Bob's age = \(1.6s\).
• From statement 3: Charlie's age = \(\frac{s + 1.6s}{2} = 1.3s\).
• Age order: Alice \((s)\) < Charlie \((1.3s)\) < Bob \((1.6s)\), confirming Alice 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: Charlie \((1.3s)\) tells Alice \((s)\): "I'm at least 5 years older." Since Charlie > Alice, Charlie 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 \(s = 10\): Alice = 10, Bob = 16, Charlie = 13.
• Statement 1 (Bob to Charlie): "Charlie is youngest" — false (Alice is youngest). Bob (16) > Charlie (13), so Bob lies. ✓
• Statement 2 (Alice to Bob): "Your age is 60% greater" — true (\(10 \times 1.6 = 16\)). Alice (10) < Bob (16), so Alice tells truth. ✓
• Statement 3 (Alice to Charlie): "Your age is the average" — true (\((10 + 16)/2 = 13\)). Alice (10) < Charlie (13), so Alice tells truth. ✓
• Statement 4 (Charlie to Alice): "At least 5 years older" — false (gap is \(13 - 10 = 3\)). Charlie (13) > Alice (10), so Charlie lies. ✓

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