# CAT - 2007

1

Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

2

In a tournament, there are n team T1 , T2 , ...., Tn , with n > 5. Each team consists of k players, k > 3. The following pairs of teams have one player in common: T& T2, T2 & T3 , ....., Tn-1 & Tn , and Tn & T1 . No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together?

3

Consider the set S = {2, 3, 4, ...., 2n + 1), where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X - Y?

4

Ten years ago, the ages of the members of a joint family of eight people added up to 231 years. Three years later, one member died at the age of 60 years and a child was born during the same year. After another three years, one more member died, again at 60, and a child was born during the same year. The current average age of this eight-member joint is nearest to

5

A function f (x) satisfies f (1) = 3600, and f (1) + f (2) + ... + f (n) = n2 f (n), for all positive integers n > 1. What is the value of f (9)?