Help with this question please...

Question

A mathematic Olympiad is attended by male and female participants of less than or equal to 2006 people.Number of male participants is more than the female.If the chance of winner and runner up champion from the same gender is 1/2,then the number of male particpants is...

A.999
B.945
C.990
D.946
E.959

Answer 1

Let m be the number of males and f be the number of females.  So the total number of attendees is m + f.

 

From this we can derive a bunch of facts:

 

1. The probability that a male wins is m / (m + f).

2. If a male wins, then there are m + f -1 other participants, of which m - 1 are male.  So the probability that a male also comes in second is (m - 1) / (m + f - 1)

3. So the probability that a male wins AND a male comes in second is [m / (m + f)] [(m - 1) / (m + f - 1)] = [m (m - 1)] / [(m + f) (m + f - 1)] = (m² - m) / [(m + f) (m + f - 1)]

4. By the same reasoning, the probability that a female wins and a female comes in second is (f² - f) / [(m + f) (m + f - 1)]

5. So the probability that either of the two events above occurs is (m² - m + f² - f ) / [(m + f) (m + f - 1)] = (m² - m + f² - f) / (m² + f² + 2mf - m - f)

 

This gives us an equation:

 

(m² - m + f² - f) / (m² + f² + 2mf - m - f) = 1/2

 

Let's see where this leads:

 

(m² - m + f² - f) = (m² + f² + 2mf - m - f) / 2

2(m² - m + f² - f) = m² + f² + 2mf - m - f

2m² - 2m + 2f² - 2f = m² + f² + 2mf - m - f

 

Bringing all the terms over to the left and simplifying, we get:

m² - 2mf + f² - m - f = 0

m² - 2mf + f² = m + f

(m - f)² = m + f

 

That tells us that the total number of people (m + f) is a perfect square.  And it is the square of the difference between the number of males and the number of females. 

 

The highest perfect square below 2006 is 1936, which is 44².  So let's suppose this is the right number of people.  That gives us two equations:

 

m + f = 1936

m - f = 44

 

Solving by elimination:

 

2m = 1980

m = 990.

 

So if there are 1936 people, there are 990 males, which is choice C.

 

But what if the total number of people is the next lower perfect square, 1849, which is 43².  That would give us the equations:

 

m + f = 1849

m - f = 43

2m = 1892

m = 946

 

That's choice D

 

What if it's the next lower perfect square, 1764, which is 42².

 

m + f = 1764

m - f = 42

2m = 1806

m = 903.

 

That's not one of the choices.  And as we go lower with perfect squares, the number of males decreases, and we're already lower than all the choices.  So we can stop there.

 

So as I see it, it can be C or D.  There doesn't seem to be anything in the problem that let's us narrow it down any further.