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# The Ultimate Maths Competition Thread

1. Does anybody have a nice solution to BMO1 2007/8 question 1? I used algebra but got stuck
2. (Original post by ben167)
Does anybody have a nice solution to BMO1 2007/8 question 1? I used algebra but got stuck
Algebra is the way to go. It should work out nicely tbh.

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3. (Original post by physicsmaths)
Algebra is the way to go. It should work out nicely tbh.

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I know but i got to a stage where it stopped working because I needed to find out 2007^4 which is huge
4. @physicsmaths how would you solve it?
5. (Original post by ben167)
I know but i got to a stage where it stopped working because I needed to find out 2007^4 which is huge
Let n=2007 then do the algebraic division to get a quadratic.

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6. Signing up to this for summer. Wanna get good at problem solving before I go to uni cause atm I'm shocking, didn't even qualify for BMO
7. a fun geo question. Latex is screwed anyway so i'm too lazy to fix it
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
8. Could anyone check that its x=0 y=1 and x=3 y=-4 as the solutions to bmo1 1987 question 1. Many thanks
9. ok another one
find the number of solutions to where and p is a prime congruent to 3 mod 4.
10. I feel like I am solving most of the geometry problems using trig is that looked down upon like calculus in olympiads?
11. (Original post by 11234)
I feel like I am solving most of the geometry problems using trig is that looked down upon like calculus in olympiads?
Nah it aint looked down upon. Its looked at better then coordinate bashes and stuff.

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12. (Original post by physicsmaths)
Nah it aint looked down upon. Its looked at better then coordinate bashes and stuff.

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But i find doing it by pure euclidean is the hardest
13. (Original post by 11234)
But i find doing it by pure euclidean is the hardest
give an example
14. (Original post by gasfxekl)
give an example
Like using constructions and similar triangles and stuff
15. (Original post by 11234)
Like using constructions and similar triangles and stuff
practice is key.
16. (Original post by gasfxekl)
practice is key.
I know do you have any geometry questions you would be happy to share. Sorry I couldnt understand the previous geometry question you posted without latex
17. Btw guys Irish Mathematical Olympiads is another massive source of questions at the exact level of BMO1/2 so you can do thos if you run out.

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18. (Original post by 11234)
I know do you have any geometry questions you would be happy to share. Sorry I couldnt understand the previous geometry question you posted without latex
lol dw that ones kinda hard
idk bmo has a lot of nice geometry questions. how much do you know about cyclic quadrilaterals etc?
19. (Original post by gasfxekl)
lol dw that ones kinda hard
idk bmo has a lot of nice geometry questions. how much do you know about cyclic quadrilaterals etc?
ptolemys theorem and that the opposite angles add to 180
20. (Original post by 11234)
ptolemys theorem and that the opposite angles add to 180
okay
some other useful stuff is angle bisector theorem and angles subtended on the same arc but im sure you know that already
okay try this one
in a quadrilateral ABCD where BC//AD the diagonals intersect in P. the circumcircles of ABP and CPD intersect AD in S and T respectively. Let the midpoint of ST be . Show that MB=MC.
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