The Ultimate Maths Competition Thread

Problem 1:

Balkan Mathematical Olympiad 2014 Question 1:

Let x, y and z be positive real numbers such that xy + yz + xz = 3xyz. Prove that

x^2y+y^2z+z^2x is greater than or equal to 2(x + y+ z) -3

and determine when the equality holds.

After putting pen to paper
Something with schurs but i don't know. Maybe cos you the x^r case etc after some work. I think it can be done this way but i won't carry on since ive solved it using a normal method.
Divide through by x,y,z to get
cylic sum 1/x=3
note that
(x+y+z-3)^2>=0
Leading to (x+y+z)^2>=6(x+y+z)-9 Call this fact (1)
now apply cauchy to LHS expression with sum of 1/x cyclic and it leads to LHS >= fact(1)/3 leading to RHS. Done


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what IeveI is this question if you don't me asking?

Problem 2:

IMO 2006:

Let ABC be a triangle with incentive I. A point P in the interior of the triangle satisfies

angle PBA + angle PCA = angle PBC + angle PCB

Show that AP is greater than or equal to AI and that equality holds if and only if P coincides with I

This problem is riciculously easy I think for IMO!
Do you do IMO problems regularly now?


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@physicsmaths
What about this?

IMO Shortlisted problem 2006:

In triangle ABC, let J be the centre of the excircle tangent to side BC at A1 and tothe extensions of sides AC and AB at B1 and C1, respectively. Suppose that the lines A1B1and AB are perpendicular and intersect at D. Let E be the foot of the perpendicular from C1to line DJ. Determine the angles ∠BEA1 and ∠AEB1.

(the numbers after the capital letters are meant to be smaller)

I haven't done this yet so I don't know I had done that Problem 1 before hand hence I knew it was easy. I can't try it right now since I have all my A level exams rn though, renzhi is currently preparing for the IMO stuff so ask him if you get stuck! Hes a badman who don't need to recise for a levels since he got straight 100s last year haha.


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Will do. Good luck in your A-Levels

Cheers, good luck in your exams too! (AS or GCSE?)


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Well I've got GCSE ICT tomorrow and AS and A2 Maths as well.

Ah so Year 11? Good luck!


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No I'm in Year 10

Oh ok. Impressive looking at IMO question in Yr 10! You will have a good shotnext year!


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@physicsmaths
@Renzhi10122
What about this?

IMO Shortlisted problem 2006:

In triangle ABC, let J be the centre of the excircle tangent to side BC at A1 and tothe extensions of sides AC and AB at B1 and C1, respectively. Suppose that the lines A1B1and AB are perpendicular and intersect at D. Let E be the foot of the perpendicular from C1to line DJ. Determine the angles ∠BEA1 and ∠AEB1.

(the numbers after the capital letters are meant to be smaller)

M8, that's a G5... I haven't tried this one before, but I'll try it now. It looks like it can be done with areals.

I think when you have the ability to do IMO questions, going through the shortlisted problems each year can be really effective because you can see where your strengths and weaknesses are depending on how many questions you solve within a specific topic (e.g. number theory)