The mean certainly will stay the same if you add a value to a set equal to the set's current mean. Because you are adding a new element which is of a distance of 0 to the mean, the standard deviation will go down (as long as the standard deviation wasn't 0 to begin with) because you will now have more elements clustered close to the mean.
It is impossible to tell what will happen to the median, however, so this is a badly designed question. Where is it from? If you have a set like the following (I'll use four elements instead of ten just for simplicity) :
1, 3, 5, 7
then inserting a new element equal to the mean, which is 4, will not change the median; the median will still be 4. However, if you have this set:
0, 2, 4, 10
then if we insert an element equal to the mean of 4, the median changes from 3 to 4.
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