Latest Maths Olympiad Aptitude Question SOLUTION: 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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Previous 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. Next Question: Define a sequence hf0(x), f1(x), f2(x), . . .i of functions by f0(x) = 1, f1(x) = x, 􀀀 fn(x) 2 − 1 = fn+1(x)fn−1(x), for n 1. Prove that each fn(x) is a polynomial with integer coefficients

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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.: Asked In Maths Olympiad MAN (10 years ago); Unsolved; Is this Puzzle helpful? (1) (1) Submit Your Solution

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Previous 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. Next Question: Define a sequence hf0(x), f1(x), f2(x), . . .i of functions by f0(x) = 1, f1(x) = x, 􀀀 fn(x) 2 − 1 = fn+1(x)fn−1(x), for n 1. Prove that each fn(x) is a polynomial with integer coefficients

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