Problem (British MO 2015) :
In Oddesdon Primary School there are an odd number of classes. Each class contains an odd number of pupils. One pupil from each class will be chosen to form the school council. Prove that the following two statements are logically equivalent.
a) There are more ways to form a school council which includes an odd number of boys than ways to form a school council which includes an odd number of girls.
b) There are an odd number of classes which contain more boys than girls.

Solution

I will prove that by induction on the number of classes.
Once this is implied, the other way is immediate (replace girls with boys)

Let the number of classes be
For , the result is trivial.

Now let
Let be the set of classes with number of boys greater than the number of girls and the set of classes with number of girls greater than the number of boys.

There are two cases

Case :
I will show that for any council with number of girls odd can be mapped to a council with number of boys odd.
Let the class in have the boy students as and girl students as
Consider a selection in which the number of girls is odd.
Let , then replacing gives and arrangement in which number of boys is odd.
And since number of girls is more than the number of boys, here we are done.
Let , due to parity, there exists some class such that a boy has been selected in this class.
Correspond each boy to a girl in this class (some girls will be left out as they are more in number). Replacing this selected boy by it's corresponding girl gives a selection with odd number of boys.

Thus this case is done.


Case :
Consider some two elements
Then is a system of classes satisfying the problem conditions.
By induction hypothesis, the number of ways to select odd number of boys () is greater than number of ways to select odd number of girls () in the council.
Let have boys and girls respectively.
Then the number of ways of selecting odd number of boys() from the classes is .
And the number of ways of selecting odd number of girls () from the classes is
Now observe that


Hence by induction, we are done.