1 I have four children. The age in years of each child is a positive integer between 2 and 16 inclusive and all four ages are distinct. A year ago the square of the age of the oldest child was equal to the sum of the squares of the ages of the other three. In one year's time the sum of the squares of the ages of the oldest and the youngest will be equal to the sum of the squares of the other two children. Decide whether this information is sufficient to determine their ages uniquely, and find all possibilities for their ages.

2 A circle has diameter AB and X is a fixed point of AB lying between A and B. A point P, distinct from A and B, lies on the circumference of the circle. Prove that, for all possible positions of P,

tanAPX divided by tanPAX

remains constant.

 

3 Determine a positive constant c such that the equation

xy² - y² - x + y = c

has precisely three solutions (x,y) in positive integers.

 

4 Any positive integer m can be written uniquely in base 3 form as a string of 0's, 1's and 2's (not beginning with a zero). For example

98 = (1x81) + (0x27) + (1x9) + (2x3) + (2x1) = (10122)₃.

Let c(m) denote the sum of the cubes of the digits of the base 3 form of m; thus, for instance

c(98) = 1³ + 0³ + 1³ + 2³ + 2³ = 18

Let n be any fixed positive integer. Define the sequence (u_r ) by

u₁ = n and u_r = c( u_r-1 ) for r „ 2.

Show that there is a positive integer r for which u_r = 1, 2 or 17.

 

5 Consider all functions f from the positive integers to the positive integers such that

(i) for each postive integer m, there is a unique positive integer n such that f(n) = m;

(ii) for each positive integer n, we have f(n+1) is either 4f(n)-1 or f(n)-1.

Find the set of positive integers p such that f(1999) = p for some function f with properties (i) and (ii).