Junior Exam J1
Each question is worth 7 marks.
Time: 4 hours
No books, notes or calculators allowed.
Note: You must prove your answer.
is defined by the rule that a term is the average of its neighbours (excluding the first term).
(a) Prove that is a term in the sequence.
(b) Find the number of times the digit occurs in the sum of all the terms in the sequence from to .
is a function defined as the product of all the factors of . e.g. .
(a) Find all such that .
(b) Find all such that .
Find all positive integral values of , and such that
Find all primes and such that
is a perfect square. Also state the perfect square.
Sets and contain positive integers such that the sum of any 2 elements in set are in set and the quotient (larger element divided by the smaller element) of any 2 elements in set are in set .
Find the maximum number of elements in .