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Author
ass0

Joined: 31 Dec 2008
Posts: 521
Location: ( . Y . )
ass0
I saw that many of you guys are so awesome in maths, so i remembered the last IMO and i thought why not do an internal contest in the fasm board, that way some of us could learn from the mathmen (I bet revolution will say 'or mathwomen').

Btw I can't resolve any of these problems xD.

http://www.imo-official.org/problems.aspx

_________________

Nombre: Aquiles Castro.
20 Oct 2009, 08:16
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17270
Location: In your JS exploiting you and your system
revolution
mathpeople? mathpersons? mathits?

BTW: With all the JS on the site there are no links to any of the problems, so I can't download to have a look-see. Are there any interesting ones?
20 Oct 2009, 09:22
ass0

Joined: 31 Dec 2008
Posts: 521
Location: ( . Y . )
ass0
Mathmen like policemen xD (the mathematicians word is boring).
Quote:

IMO 2009 Germany - Questions

Wednesday, July 15, 2009

Problem 1. Let n be a positive integer and let a1,a2,a3,...,ak ( k≥ 2) be distinct integer in the set { 1,2,...,n} such that n divides ai(ai+1-1) for i = 1,2,...,k-1. Prove that n does not divide ak(a1-1).

Problem 2. Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB respectively. Let K,L and M be the midpoints of the segments BP,CQ and PQ. respectively, and let Γ be the circle passing through K,L and M. Suppose that the line PQ is tangent to the circle Γ. Prove that OP = OQ.

Problem 3. Suppose that s1,s2,s3, ... is a strictly increasing sequence of positive integers such that the sub-sequences ss_{1},ss_{2},ss_{3}, ... and ss_{1+1},ss_{2+1},ss_{3+1}, ... are both arithmetic progressions. Prove that the sequence s1,s2,s3, ... is itself an arithmetic progression.

Language: English Time: 4 hours and 30 minutes
Each problem is worth 7 points

Quote:

Thursday, July 16, 2009

Problem 4. Let ABC be a triangle with AB = AC . The angle bisectors of ∠ C AB and ∠ AB C meet the sides B C and C A at D and E , respectively. Let K be the incentre of triangle ADC. Suppose that ∠ BEK = 45ยบ . Find all possible values of ∠ C AB .

Problem 5. Determine all functions f from the set of positive integers to the set of positive integers such that, for all positive integers a and b, there exists a non-degenerate triangle with sides of lengths
a, f(b) and f(b+f(a)-1).
(A triangle is non-degenerate if its vertices are not collinear.)

Problem 6. Let a1, a2, ... , an be distinct positive integers and let M be a set of n-1 positive integers not containing s = a1+a2+ ...+an. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1, a2, ... , an in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.

Language: English Time: 4 hours and 30 minutes
Each problem is worth 7 points

_________________

Nombre: Aquiles Castro.

Last edited by ass0 on 20 Oct 2009, 11:26; edited 1 time in total
20 Oct 2009, 09:41
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17270
Location: In your JS exploiting you and your system
revolution
Reading those problems reminds me why I am not a mathwhatsit.

Ouch, my brain hurts.
20 Oct 2009, 09:49
ass0

Joined: 31 Dec 2008
Posts: 521
Location: ( . Y . )
ass0
...and those are for teenagers (High School students). But what surprises me is that Chinese kids rock so hard in those Olympiads.

http://www.imo-official.org/results.aspx
http://www.imo-official.org/team_r.aspx?code=CHN&year=2009
20 Oct 2009, 10:01
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17270
Location: In your JS exploiting you and your system
revolution
Doesn't surprise me at all. I have met a lot of Chinese people and I can understand why, as a group, they would do well.
20 Oct 2009, 10:05
ass0

Joined: 31 Dec 2008
Posts: 521
Location: ( . Y . )
ass0
It makes me scared.

_________________

Nombre: Aquiles Castro.
20 Oct 2009, 10:10
r22

Joined: 27 Dec 2004
Posts: 805
r22
Cultural tendencies/leanings have such a strong affect on the development of children.
On an unrelated note I just noticed ass0's location
Quote:
Joined: 31 Dec 2008
Posts: 176
Location: ( . Y . )

I suppose, when your nationality makes up 17% (Chinese) of the worlds population there's a lot of extra pressure to stand-out.

With the power of the Internet at my disposal I'd likely only be able to answer 2 (3 tops) of those 6 questions listed in 4 1/2 hours.
20 Oct 2009, 17:58
kohlrak

Joined: 21 Jul 2006
Posts: 1421
Location: Uncle Sam's Pad
kohlrak
hm.... Tomasz....
20 Oct 2009, 22:33
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