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# The Ultimate Maths Competition Thread

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1. This thread was created for the people who are interested in discussing and sharing problems relating to mathematical competitions. Hopefully, this thread can act as a place where people can talk about problems ranging from the UK Maths Challenges all the way to IMO-standard problems and share resources so people can improve their ability at solving mathematical questions in competitions. Also, I hope that this thread can help people to develop their abilities and interests in problem solving and find new people who share an interest in maths and problem solving.

Happy solving.

Renzhi10122
physicsmaths

(I have tagged two people who are more popular TSR members and problem solving enthusiasts so that this thread can gain in popularity and reach as many people as possible)
2. 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.
3. (Original post by Maths465Man)
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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4. (Original post by physicsmaths)
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?
5. (Original post by Yunique)
what IeveI is this question if you don't me asking?
Below IMO but higher then National Maths Competitions.

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6. 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
7. (Original post by Maths465Man)
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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8. (Original post by physicsmaths)
This problem is riciculously easy I think for IMO!
Do you do IMO problems regularly now?

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I try however I can do very few of them.
9. physicsmaths
Renzhi10122

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)
10. (Original post by Maths465Man)
physicsmaths

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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11. (Original post by physicsmaths)
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
12. (Original post by Maths465Man)
Will do. Good luck in your A-Levels
Cheers, good luck in your exams too! (AS or GCSE?)

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13. (Original post by physicsmaths)
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.
14. (Original post by Maths465Man)
Well I've got GCSE ICT tomorrow and AS and A2 Maths as well.
Ah so Year 11? Good luck!

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15. (Original post by physicsmaths)
Ah so Year 11? Good luck!

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No I'm in Year 10
16. (Original post by Maths465Man)
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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17. (Original post by physicsmaths)
Oh ok. Impressive looking at IMO question in Yr 10! You will have a good shotnext year!

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I'd love to make it to an IMO, but it will be incredibly difficult.
18. (Original post by Maths465Man)
physicsmaths
Renzhi10122

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.
19. (Original post by Renzhi10122)
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)
20. (Original post by Maths465Man)
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)
K, turns out that the G5 wasn't that hard
And yes, I agree, that's what I've been doing for a while now, and it's a very good source of questions. Unfortunately, questions start getting rather hard the higher the shortlist number...
Spoiler:
Show
So let F be the intersection of CJ and A1B1. Then you have a rectangle BC1JF. (JA1)^2=JC*JF, because of circle CA1F being tangent to JA1, then looking at the rectangle, that implies JC/JC1=JC1/DC1, so JC1C and C1DJ are similar, and it follows that DJ is perp. to C1C. Then J,E,A1,C,B1 are concyclic. Also, A=B, so then both BDA1E and ADEB1 are cyclic, hence the two angles that are wanted are both 90.

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