Olympiad Math 2008

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-----SECTION A------
Question 1
Given
(1 + 2 + 3 + .... + n) + k = 202
Find k.

I dont know how to solve, but my friend get k = 12.

Question 2
Forgotten


Question 3
ABCD is a cyclic quadrilateral. Given angle A : angle B : angle C = 2007 : 2008 : 2009,
Find angle B.


My answer:
Cyclic quadrilateral -> angle A + angle C = angle B + angle D = 180
angle A / angle C = 2007/2009
angle A
= 2007 / (2009 + 2007) x 180
= 2007/ 4016 x 180 <- retain this.

angle B / angle A = 2008 / 2007
angle B
= 2008/2007 x angle A
= (2008/2007) x (2007/4016) x 180 <- cancel each other (calculator is not allowed)
= 90

Question 4
ABCD is a square with length 1. M is the midpoint of AB and N is the midpoint of BC. The intersection of line DM and AN is marked as P. Find the area of triangle AMP

Dont know how to solve.

Question 5
a₂ - a₁ = a₃ - a₂ = a₄ - a₃ = a₅ - a₄ = 30
Given all a's are prime numbers and a₅ < 150,
find the minimum value of a.

I not sure whether got other method.
I use substituition.
a₁ = 7.

Question 6
Solve
2008² - 2007² + 2006² - 2005² + ..... + 2² - 1²

I get this wrong but eventually know how to solve.

Recall x² - y² = (x + y)(x - y)
(2008-2007)(2008+2007) + (2006-2005)(2006+2005) + ...... + (2-1)(2+1)
= (1)(2008+2007) + (1)(2006+2005) + ....... + (2+1)
= 2008 + 2007 + 2006 + 2005 +...... + 2 + 1

Using formula n(n+1) divide by 2,
= [2008 (2009)] / 2
= 1004 x 2009
= 2017036

-----SECTION B------
Question 1
Show that
a² + b² + c² + ab + ac + bc + 6 equal or greater than 4 (a + b + c)

I dont know how to do.

Question 2
(number in degree)
ABC is an isoceles triangle where angle A = angle B = 80.
J lies on AC where angle ABJ = 10.
K lies on AB where angle ACK = 20.
M intersects line BC, where BM = MC.
The intersection of line AM and CK is N.
SHOW THAT line JN is parallel to line AB.

Complicated. I gave up.

Question 3
Slimplify
[ 1 + n (n+1) (n+2) (n+3)] ^1/2

= [ 1 + (n+2) (n+3) n² + 3n ] ^1/2
= [ 1 + (n+2) (n+3) n² + 3n + 2 - 2 ] ^1/2
= [ 1 + (n+2) (n+3) (n+2) (n+3) - 2 ] ^1/2
= [(n+2)² (n+3)² -1 ] ^1/2
= (n+2)(n+3)-1

Please verify this question whether I got correct or wrong.

Question 1
(1+2+3+...+n)< 202

Found that (1+2+....+19) = 190 (If add 20 will be 210)

Since sum should be less than 202, then 202-190=12 k=12

Thanks a lot! I understand it now.

Section B question 1

Well, it just asks you to 'show' that overall > or = to 4(a+b+c)

Substitute all unknowns with 1,2 or whatever . Result obtained will be more than or equal.

Exp :
Substitute a=b=c=1

(1+1+1+1+1+1)+6 equal or greater than 4(1+1+1)

12 equal to 12

Solution above is just my thought. Btw where did you get these? It seems more interesting than Add Math.

Actually I went for Olympaid Math this year. I am really stupid and don't know how to do most of he question, yet the teacher chosen me... haiz, no choice.

bobby1, you seem to be good at math!
Thanks for answering those questions!

You can answer some of the questions, it's not bad already. These questions are more of critical thinking than same format like our math and add math exam.
You must be quite good to be chosen as candidate. I wanna go too but my school never takes part in such event. Now I'm already Form 5 and will be sitting for SPM. I'm curious about it. :/ That's why I went here for Trial paper.
Well, I always think our math syllabus is like a crap compare to foreign countries. Add Math has a little bit challenge once you understand the whole concepts.

Aha, I love add math, but I am very very bad at critical thinking. T.T

Just think out of the range in our syllabuses. Eventually you would be able to figure it out. Study more and develop your own thinking. It's like Albert Einstein studies and develop his infamous E=mc^2 formula.

Try section A question 4 once again im here to guide.
convert it into a graphical question with x and y axis labelled.

Try section B question 2 too

This post has been edited by Desmond on Dec 28 2008, 01:53 AM