Hi Sir,


The following are my workings, please comment:

2[3^(6n)] + k[2^(3n+1)] - 1
=2[9^(3n)] + 2k(8^n) - 1

Let 7 = x - 1
=> f(x) = 2[(x+1)^(3n)] + 2k(x^n) - 1
=> f(1) = 2^(3n+1) + 2k - 1 = 0
k = 1/2 - 2^(3n) -> -ve? :?

Yo Sir,


Please help:
Find the number of positive integers k<100 such that 2[3^(6n)] + k[2^(3n+1)] - 1 is divisible by 7 for any positive integer n.

Thanks

Hi Sir,


Thanks. I think the remainders have to be the same, so that the quotient is common, in this case is x (qoute from the 高人 (CoffeCat) from the previous question: If the number has the same remainder 1 upon division by 3 and 5, it will leave the same remainder 1 upon division by 15 -> 15x+1 ).
By long division or algebraic juggling, (1-11x-4x^2+3x^3-2x^4)/(6-4x+3x^2-2x^3) = x + (1-17x)/(6-4x+3x^2-2x^3).
(1-17x)/(6-4x+3x^2-2x^3) = (1-17x)/[(3-2x )(x^2 + 2)] = A/(3-2x) + (Bx+C)(x^2 + 2).

Hi CoffeeCat,


paiseh, from the questions asked, 不值一提 :oops:.

Hi CoffeCat,


Ya, currently in school Math Talent Program training. Find a wealth of resources here. 请多多指教. Thanks.

Hi Coffeecat,


Thanks, got it. Cheers. 卧虎藏龙 here

Hi Sir,


Thanks.

Please help to resolve the fraction (1-11x-4x^2+3x^3-2x^4)/(6-4x+3x^2-2x^3) into partial fraction.

Thanks.

Hi Sir,


Thanks a lot, but for Q3 need some time to digest.

Hi Sir,


Thanks to all, excellent .

A few more headache questions :? :
1. Two vertical poles, 20m and 80m high, stand apart on a horizontal plane. The height, in metres, of the point of intersection of the lines joining the top of each pole to the foot of the other is .....

2. Some unit cubes are assembled to form a larger cube and then some of the faces of these larger cubes are painted. After the paint dries, the larger cubes is disassembled into the unit cubes, and it is found that 45 of these have no paint on any of their faces. How many faces of the larger cube were painted?

3. In how many different ways can a careless office boy place four letters in four envelopes so that no one gets the right letter?

Thanks.

Hi Sir,


Need your help for some maths competition questions:

1. When 3^1981 + 2 is divided by 11, the remainder is …
2. The least positive integer which has remainders 1, 1 and 5 when divided by 3, 5 and 7 respectively, is (A)166 (B)151 145 (D)131 (E)none of these.
3. If x and y are integers such that (x-y)² + 2y² = 27, then the only number x can be are (A)3,5 (B)-6,4 0,4,6 (D)0,-4,4,-6,6 (E)0,-2,2,-4, 4,-6,6.

Thanks.