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:

https://olympiads.hbcse.tifr.res.in

(I am a tutor for such mathematical olympiads).

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.