Pre RMO more algebra problems for practice

Question I:

I rode one third of a journey at 10kmph, one third more at 9, and the rest at 8 kmph; if I had ridden half the journey at 10kmph, and the other half at 8 kmph, I should have been half a minute longer on the way: what distance did I ride?

Question 2:

The express train leaves Bristol at 3pm and reaches London at 6pm; the ordinary train leaves London at 1:30pm and arrives at Bristol at 6pm. If both trains travel uniformly, find the time when they will meet.

Question 3:

Solve (a) 0.\dot{6}x + 0.75x-0.1\dot{6} = x - 0.58\dot{3}x+5

Solve (b) \frac{37}{x^{2}-5x+6} + \frac{4}{x-2} = \frac{7}{3-x}

Question 4:

Simplify: (1+x)^{2} \div \{ 1 + \frac{x}{1-x+ \frac{x}{1+x+x^{2}}}\}

Question 5:

Find the square root of \frac{4a^{2}-12ab-6bc+4ac+9b^{2}+c^{2}}{4a^{2}+9c^{2}-12ac}

Question 6:

Find the square root of 4a^{4}+9(a^{2}+\frac{1}{a^{2}})+12a(a^{2}+1)+18

Question 7:

Solve the following system of equations:

\frac{1}{3}(x+y)+2z=21

3x - \frac{1}{2}(y+z) = 65

x + \frac{1}{2}(x+y-z) = 38

Question 8:

A number consists of three digits, the right hand one being zero. If the left hand and middle digits be interchanged the number is diminished by 180; if the left hand digit be halved and the other two digits are interchanged, the number is diminished by 336; find the number.

Question 9:

Add together the following fractions:

\frac{2}{x^{2}+xy+y^{2}}, \frac{-4x}{x^{3}-y^{3}}, \frac{x^{2}}{y^{2}(x-y)^{2}}, and \frac{-x^{2}}{x^{3}y-y^{4}}

Question 10:

Simplify:

\frac{a^{3}+b^{3}}{a^{4}-b^{4}} - \frac{a+b}{a^{2}-b^{2}} -\frac{1}{2} \{ \frac{a-b}{a^{2}+b^{2}} - \frac{1}{a-b} \}

More later,
Nalin Pithwa

Pre RMO Algebra problems for practice

Question 1:

Find the square root of 49x^{4}+\frac{1051x^{2}}{25} - \frac{14x^{3}}{5} - \frac{6x}{5} + 9

Question 2:

The surface area of a circular cone is given by A= {\pi}r^{2}+{\pi}rs, where s cm is the slant height, r cm is the radius of the base and \pi is \frac{22}{7}. Find the radius of the base if a cone of surface area 93.5 square cm has a slant height of 5 cm.

Question 3:

Solve:

\frac{a+x}{a^{2}+ax+x^{2}} +\frac{a-x}{a^{2}-ax+ x^{2}} = \frac{3a}{x(a^{4}+a^{2}x^{2}+x^{4})}

Question 4:

Subtract \frac{x+3}{x^{2}+x-12} + \frac{x+4}{x^{2}-x-12} and divide the difference by 1 + \frac{2(x^{2}-12)}{x^{2}+7x+12}

Question 5:

Solve the following for the unknown x:

\frac{x}{2(x+3)} - \frac{53}{24} = \frac{x^{2}}{x^{2}-9} - \frac{8x-1}{4(x-3)}

Question 6:

Find the square root of the following:

a^{6}+ \frac{1}{a^{6}} -6(a^{4}+\frac{1}{a^{4}}) +12(a^{2}+\frac{1}{a^{2}})-20; also, cube the result.

More later,
Nalin Pithwa.