National Mathematics Olympiad.

This morning I just went to MMU to sit for that.
Below are the questions as I remembered (sorry if the is any mistake in it)

Part A (Only final answer needed)

1. If a(1)=6,..... a(n)=6^(a(n-1)) find the remainder when a(100) is divided by 11.
2. Given y is bigger than -1 but smaller than 0 and x is bigger than 0 but smaller than 1 and A="yx^2" B="(yx^2)^(-1)" C="xy^2" D="(xy^2)^(-1)" arrange from smallest to biggest.
   
3. A triangle has one 4cm side, the other 2 sides are in the ratio of 1:3 find the maximum area of the triangle.
4. Simplify log(base2)4 log(base4)6log(base6)8......log(base2n)(2n+2)
5. Express 580 as the sum of 2 squares.
6. f is a function defined for non-negative integers and f(2n+1)=f(n) f(2n)=1-f(n) find f(2007).

Part B (Show full working)

1. f and g are functions defined for [0,2c] where c>0 ,prove that there exist (x,y) which is an element of [0,2c] such that {modulus(xy-f(x)+g(y))} >or= to c^2
2. Two circles of radius 1 and 2 respectively are touched tangentially on the external. A third circle touches both of the tangentially, making the centre of the circles form a right angle triangle. Find the radius of the third circle.
3. m,n is a element of the set {1,2,3,4,.......2007} Find the maximum value of m^2+n^2 if (n^2-mn-m^2)^2=1

The time is 2hours and 30 minutes and no calculator is allowed.


I did all of them.
Try them out yourself. If you want my answers or you don't understand what I wrote, leave a comment or message me.




Now I just checked my SAT2 scores and they are
Physics 800
Maths level2 800
Chemistry 790
with 800 as the maximum score, it's not too bad right?
Ok that's all for now until next time, Cheers.