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THE SOLUTION

A`s statement must be true; if he is lying, he is last and he is also first or second, which is contradictory. So A is third or fourth. If D is telling the truth, E is lying and D is first and is lying, which is contradictory. So E is not second, and D is first or second. If E is lying, D is first and E is second. But D is lying when he says E is second. Therefore, E is telling the truth, which makes him third, fourth or fifth. D is not the first, but second. Only B or C can be first. If B is not first, since he is also not second he is telling the truth, and C is third, and cannot be first either, which is impossible. Then B is first and C is not third. C, being neither first nor second, is telling the truth that A is behind E. E is third, A is fourth, and C is fifth. The athletes places in order; B,D,E,A,C.