Solution: Day 2, problem 1
The units form a cyclic group <a | a5 = 1>. Since -1 is a unit of order dividing 2, it equals 1, so 2 = 0. A computation then shows that the element x = 1 + a2 + a3 has cube equal to 1.
Since 3 is prime to 5, we must have x = 1, i.e. a2(a + 1) = 0. But then a + 1 = 0, which is a contradiction.
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Don Zagier 1996