# Pre RMO algebra : some tough problems

Question 1:

Find the cube root of $x^{3} -12x^{2} + 54x -112 + \frac{108}{x} - \frac{48}{x^{2}} + \frac{8}{x^{3}}$

Question 2:

Find the square root of $\frac{x}{y} + \frac{y}{x} +3 - 2\sqrt{\frac{x}{y}} -2\sqrt{\frac{y}{x}}$

Question 3:

Simplify (a):

$(\frac{x}{x-1} - \frac{1}{x+1}). \frac{x^{3}-1}{x^{6}+1}.\frac{(x-1)^{2}(x+1)^{2}+x^{2}}{x^{4}+x^{2}+1}$

Simplify (b):
$\{ \frac{a^{4}-y^{4}}{a^{2}-2ay+y^{2}} \div \frac{a^{2}+ay}{a-y} \} \times \{ \frac{a^{5}-a^{3}y^{2}}{a^{3}+y^{3}} \div \frac{a^{4}-2a^{3}y+a^{2}y^{2}}{a^{2}-ay+y^{2}}\}$

Question 4:

Solve : $\frac{3x}{11} + \frac{25}{x+4} = \frac{1}{3} (x+5)$

Question 5:

Solve the following simultaneous equations:

$2x^{2}-3y^{2}=23$ and $2xy - 3y^{2}=3$

Question 6:

Simplify (a):

$\frac{1- \frac{a^{2}}{(x+a)^{2}}}{(x+a)(x-a)} \div \frac{x(x+2a)}{(x^{2}-a^{2})(x+a)^{2}}$

Simplify (b):

$\frac{6x^{2}y^{2}}{m+n} \div \{\frac{3(m-n)x}{7(r+s)} \div \{ \frac{4(r-s)}{21xy^{2}} \div \frac{(r^{2}-s^{2})}{4(m^{2}-n^{2})}\} \}$

Question 7:

Find the HCF and LCM of the following algebraic expressions:

$20x^{4}+x^{2}-1$ and $25x^{4}+5x^{3} - x - 1$ and $25x^{4} -10x^{2} +1$

Question 8:

Simplify the following using two different approaches:

$\frac{5}{6- \frac{5}{6- \frac{5}{6-x}}} = x$

Question 9:

Solve the following simultaneous equations:

Slatex x^{2}y^{2} + 192 = 28xy\$ and $x+y=8$

Question 10:

If a, b, c are in HP, then show that

$(\frac{3}{a} + \frac{3}{b} - \frac{2}{c})(\frac{3}{c} + \frac{3}{b} - \frac{2}{a})+ \frac{9}{b^{2}}=\frac{25}{ac}$

Question 11:

if $a+b+c+d=2s$, prove that

$4(ab+cd)^{2} - (a^{2}+b^{2}-c^{2}-d^{2})^{2}= 16(s-a)(s-b)(s-c)(s-d)$

Question 12:

Determine the ratio $x:y:z$ if we know that

$\frac{x+z}{y} = \frac{z}{x} = \frac{x}{z-y}$

More later,
Nalin Pithwa

Those interested in such mathematical olympiads should refer to: