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the King’s Factor Year 12 further questions 3 1. [2010 STEP I question 1] Given that 5x2 + 2y 2 − 6xy + 4x − 4y ≡ a(x − y + 2)2 + b(cx + y)2 + d find values of the constants a, b, c and d. Solve the simultaneous equations 5x2 + 2y 2 − 6xy + 4x − 4y = 9 6x2 + 3y 2 − 8xy + 8x − 8y = 14. 2. [2008 STEP I question 3] Prove that, if c ≥ a and d ≥ b, then ab + cd ≥ bc + ad. (?) (i.) If x ≥ y use (?) to show that x2 + y 2 ≥ 2xy. If, further, x ≥ z and y ≥ z, use (?) to show that z 2 + xy ≥ xz + yz and deduce that x2 + y 2 + z 2 ≥ xy + yz + zx. Prove that the inequality x2 + y 2 + z 2 ≥ xy + yz + zx holds for all x, y and z. (ii.) Show similarly that the inequality s t r + + ≥3 t r s holds for all positive r, s and t. 3. [Inspired by ‘Problem Solving in Recreational Mathematics’ by Averbach and Chein] The notation a mod b = c (“a modulo b equals c”), where a, b and c are integers, means that a = kb + c for some integer k and 0 ≤ c < b. Hence if a mod b = 0 then a is a multiple of b. (i) Prove that N ≡ ax + by is divisible by n if a mod n = 0 and b mod n = 0, where x, y, a, b are all integers. Now show that (a × b) mod c = (a mod c) × (b mod c). (ii) The digital sum of an integer, N , denoted d(N ) is the sum of the digits used to write the number, e.g. d(143) = 1 + 4 + 3 = 8 or d(40673) = 4 + 0 + 6 + 7 + 3 = 20. Prove that if an integer N is divisible by 9, then its digital sum is also divisible by 9. Explain why a similar statement is also true for numbers which are divisible by 3. (iii) The alternating digital sum a(N ) of an integer N is similar to the digital sum except that instead of adding all the digits of a number, the first digit is added while the second digit is subtracted, the third is added, the fourth is subtracted and so on (i.e. the sign of the terms in the sum alternates), e.g. a(143) = 1−4+3 = 0, a(40673) = 4−0+6−7+3 = 6. Prove that any integer, N , which is divisible by 11 has an alternating sum, a(N ), which is also divisible by 11. (iv) Let N = 123a5678b, where a and b denote unknown digits in the number N . Find integers a and b such that N mod (99) = 0. Hence explain how you could start to write down 1440 (or more!) other numbers which are divisible by 99 without any further computation. 1