Colour of bear in Earth's surface A bear stands in one point of the Earth's surface. Walking one kilometer south , then walking one kilometer east and immediately after one kilometer north and reaches the point from which started.Find the color of the bear. This is a question in a very old mathematical competition.I am confused because i can not think any mathematical way to approach it.Any ideas would be appreciated.

There are infinitely many solutions to this problem, all of which place the starting point either at the north pole, or very close to the south pole. Hence the bear "must" be white. Of course, real polar bears only live north of the Arctic circle, and they like to be near water, so they won't be anywhere near the north pole. Hence the puzzle is biologically incorrect.

North pole: The circle exactly one mile away from the north pole has all its points one mile south of the north pole. The eastward walk is along this circle.

Infinite family: If the starting point is a bit more than a mile north of the south pole, then the one mile east will walk exactly $n$ circles around the south pole. (by varying $n$ you can get infinitely many solutions)