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Re: 2012 Year 9 &10 Mathematics Marathon hate these kinds of questions & latex -.- $i. In $ \triangle $ OCB:\\ OC = r (given)\\$\angle$ OBC = 30$^{\circ}$ (half of 60$^{\circ}$)\\\\sin 30$^{\circ} = \frac{OC}{OB}\\\frac{1}{2} = \frac{OC}{OB}\\ \therefore$ OB = $2r$ (given OC = r)\\\\ii. In $...

Re: 2012 Year 9 &10 Mathematics Marathon \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{xz+z+1}\\$Converting the denominator to $yz + y + 1$ by using $ xyz=1 $ (e.g. $x = \frac{1}{yz})\\=\frac{yz(x+1)}{yz+y+1} + \frac{(y+1)}{yz+y+1} + \frac{xy^2z(z+1)}{yz+y+1}\\=\frac{xyz + yz + y+ 1 +...

Re: 2012 Year 9 &10 Mathematics Marathon Simplify the following algebraic fractions: $1. $\left ( 1 + \frac{45}{x-8} -\frac{26}{x-6} \right ) \left ( 3 - \frac{65}{x+7} +\frac{8}{x-2} \right )\\ $2. $\left ( 2 - \frac{3n}{m} +\frac{9n^2-2m^2}{m^2+2mn} \right ) \div \left (...

Re: 2012 Year 9 &10 Mathematics Marathon \frac{1}{(a-b)(a-c)} + \frac{1}{(b-a)(b-c)} + \frac{1}{(c-a)(c-b)}\\ = \frac{1}{(a-b)(a-c)} + \frac{1}{-(a-b)(b-c)} + \frac{1}{(a-c)(b-c)}\\ = \frac{(b-c)-(a-c)+(a-b)}{(a-b)(a-c)(b-c)}\\ = \frac{0}{(a-b)(a-c)(b-c)}\\ = 0

Re: 2012 Year 9 &10 Mathematics Marathon (x+y\sqrt{d})^2 (x-y\sqrt{d})^2\\= (x^2+2xy\sqrt{d}+y^2d)(x^2 - 2xy\sqrt{d}+y^2d)\\ = [(x^2+y^2d)+2xy\sqrt{d}][(x^2+y^2d)-2xy\sqrt{d}]\\={(x^2+y^2d})}^2 - {(2xy\sqrt{d})}^2

Re: 2012 Year 9 &10 Mathematics Marathon thanks for the advice :)

Re: 2012 Year 9 &10 Mathematics Marathon a^5-a^4b\geq ab^4-b^5\\a^5-a^4b - ab^4+b^5\geq 0 \\ a^4 (a-b) - b^4 (a-b) \geq 0 \\ (a^4-b^4)(a-b)\geq 0\\ (a^2+b^2)(a^2-b^2)(a-b)\geq 0 \\(a^2+b^2)(a+b)(a-b)(a-b) \geq 0 \\ (a^2+b^2)(a+b)(a-b)^2 \geq 0 \\\\$ Since a and b are positive, and all squares...

Re: 2012 Year 9 &10 Mathematics Marathon 16^x - 18(4^x) + 32 = 0\\ 4^{2x} - 18 \times 4^x + 32 = 0\\ 4^x (4^2 - 18) + 32 = 0\\ 4^x \times -2 + 32 = 0\\4^x\times -2=-32\\4^x = 16\\4^x = 4^2\\ \therefore x = 2

Re: 2012 Year 9 &10 Mathematics Marathon $i. $ n = 2k + 1 $ or $ n = 2k - 1\\\\$ii. Substituting $n = 2k + 1$ into the equation $ n^2+4n-1 \\(2k + 1)^2 + 4 (2k+1) - 1\\= 4k^2 + 4k + 1 + 8k + 4 - 1\\= 4k^2 + 12k + 4\\$Since everything are multiples of 4$\\\therefore n^2 + 4n - 1 $ is...

Re: 2012 Year 9 &10 Mathematics Marathon 1 :)

Re: 2012 Year 9 &10 Mathematics Marathon \frac{1}{9}$?$

Re: 2012 Year 9 &10 Mathematics Marathon $i. $\left ( x-3 \right )^{2}-6 $ii. $x = \ 3\pm \sqrt{y+6} $iii. All real y except y $<$ -6$

Re: 2012 Year 9 &10 Mathematics Marathon \frac{2}{xy}

Re: 2012 Year 9 &10 Mathematics Marathon 5 kg ?

Re: 2012 Year 9 &10 Mathematics Marathon probably 96 m2 ?

Re: 2012 Year 9 &10 Mathematics Marathon (\sqrt{x}+\sqrt{6})(\sqrt{x}-\sqrt{6})

Re: 2012 Year 9 &10 Mathematics Marathon 0.577 to 3 dp ?