Maths Olympiad
Exam
Let p1 < p2 < p3 < p4 and q1 < q2 < q3 < q4 be two sets of prime numbers such that p4 + p1 = 8 and q4 + q1 = 8. Suppose p1 > 5 and q1 > 5. prove that 30 divides p1 + q1.
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Maths Olympiad Other Question
Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides
b, b divides c and n = a + b + c.
(i) Show that all but a finite number of natural numbers are faithful.
(ii) Find the sum of all natural numbers which are not faithful.
Define a sequence hf0(x), f1(x), f2(x), . . .i of functions by
f0(x) = 1, f1(x) = x,
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fn(x)
2
− 1 = fn+1(x)fn−1(x), for n 1.
Prove that each fn(x) is a polynomial with integer coefficients