What is the largest three digit number with the

26. What is the largest three-digit number with the property that the number is equal
to the sum of its hundreds digit, the square of its tens digit and the cube of its units
digit?
27. Igor wants to make a secret code of five-letter words. To make them easy to say, he
follows these two rules:
(i) no more than two consonants or two vowels in succession
(ii) no word to start or end with two consonants
He rejects the letter ‘Q’ as too hard, so he has 20 consonants and 5 vowels to choose
from. If N is the number of code words possible, what are the first three digits of N ?
28. Consider the sequence a
1
, a
2
, a
3
, a
4
, . . . such that a
1
= 2 and for every positive
integer n ,
a
n +1
= a
n
+ p
n
, where p
n
is the largest prime factor of a
n
.
The first few terms of the sequence are 2 , 4 , 6 , 9 , 12 , 15 , 20. What is the largest value
of n such that a
n
is a four-digit number?
29. A lattice point in the plane is a point whose coordinates are both integers. Consider
a triangle whose vertices are lattice points (0 , 0), ( a, 0), and (0 , b ), where a ≥ b >
0. Suppose that the triangle contains exactly 74 lattice points in its interior, not
including those lattice points on the sides of the triangle. Determine the sum of the
areas of all such triangles.
30. A polynomial p ( x ) is called self-centered if it has integer coefficients and p (100) = 100.
If p ( x ) is a self-centred polynomial, what is the maximum number of integer solutions
k to the equation p ( k ) = k
3
?