Download the King’s Factor Year 12 further questions 3

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.
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