Others Maths Puzzle

When a two-digit number is increased by 4, the sum of its digits is equal to half
of the sum of the digits of the original number. How many possible values are
there for such a two-digit number?

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Others Other Question
The sum of 1006 different positive integers is 1019057. If none of them is
greater than 2012, what is the minimum number of these integers which must be
odd? In a village, friendship among girls is mutual. Each girl has either exactly one
friend or exactly two friends among themselves. One morning, all girls with two
friends wear red hats and the other girls all wear blue hats. It turns out that any
two friends wear hats of different colours. In the afternoon, 10 girls change their
red hats into blue hats and 12 girls change their blue hats into red hats. Now it
turns out that any two friends wear hats of the same colour. How many girls are
there in the village? (A girl can only change her hat once.)