a) The claim can be transferred into this way : think about that point “0” on the circle has a temperature T(0) and point “兀” has a temperature T(兀). So it is obvious that we only have three situations: 1) T(0)=T(兀) 2) T(0)<T(兀) 3) T(0)>T(兀) . And now let’s discuss them.

1) if T(0)=T(兀), that means point “0” and its antipodal point “兀” has the same temperature,the claim is true

2）if T(0)<T(兀), let’s think about a smooth curve f(x)=T(x)-T(x+兀) so it is obvious that f(0)<0 and f(兀)>0(f(2兀)=f(0) for it is a circle), for f(x) is a continuous function, that is, f(x) can be every value between f(0) and f(兀). And since f(0)<0 and f(兀)>0 , 0 is included in the interval (f(0),f(兀)). We can find that there must be some value a such that f(a)=0. Otherwise , that means f(x) is disconnected in 0, and thus f(x) is discontinuous which is contradictory.

3) the same as the second situation.

so to draw a conclusion there must be a such that T(a)=T(a+兀) on a circle

b) My explanation rely heavily on continuity