Five competitors -- A, B, C, D, and E -- enter a swimming race that awards gold, silver, and bronze medals to the first three to complete it. Each of the following compound statements about the race is false, although one of the two clauses in each may be true. A didn't win the gold, and B didn't win the silver. D didn't win the silver, and E didn't win the bronze. C won a medal, and D didn't. A won a medal, and C didn't. D and E both won medals. Who won each of the medals?

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A wins gold, D wins silver, C wins bronze.

Explanation:

Statement one: A didn't win the gold, and B didn't win the silver.
Statement two: D didn't win the silver, and E didn't win the bronze.
Statement three: C won a medal, and D didn't.
Statement four: A won a medal, and C didn't.
Statement five: D and E both won medals.

Look at the third statement. We have three scenarios: C doesn't win a medal and D doesn't win a medal (first clause false); C wins a medal and D wins a medal (second clause false); C doesn't win a medal and D wins a medal (both clauses false).

We cannot have a statement that is completely true so:
1. It is impossible for C to have won a medal while D didn't
2. If one clause is true, the other MUST be false

(*)If D wins a medal, we must conclude that E didn't win a medal (statement five). That must mean E didn't win bronze, meaning D must have won silver (statement two). If D won silver, B couldn't have won a silver, meaning that A won gold (statement one).

(**)If C doesn't win a medal, we must conclude A doesn't win a medal (statement four). That must mean that A doesn't win gold, so B must have won silver (statement one). If B wins silver, then D couldn't have won silver, which means E must have won bronze (statement two).

If D doesn't win a medal, then D doesn't win silver, meaning that E must have won bronze (statement two). That also means that C did not win a medal (statement three), so we repeat what was stated in paragraph (**): A doesn't win a medal, B wins silver, E wins bronze.

If C wins a medal, then D wins a medal (statement three). This means we repeat what was stated in paragraph (*): E didn't win a medal, D wins silver, and A wins gold.

Now let's look at scenario 1, where C doesn't win a medal and D doesn't win a medal. Therefore, A doesn't win a medal. But there can only be two competitors who don't win medals, so scenario 1 is out.

Now let's look at scenario 3, where C doesn't win a medal and D wins a medal. Therefore, A doesn't win a medal, A wins gold, B wins silver, D wins silver, E wins bronze, and E doesn't win a medal. Well that's full of contradictions so scenario 3 is out.

That leaves us with scenario 2, where C wins a medal and D wins a medal. Therefore, A wins gold, B doesn't win silver, D wins silver, and E doesn't win a medal. Since A wins gold and D wins silver, bronze is the only medal left to be awarded. C wins a medal, so C gets bronze.

No one won

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