# IMO 2011 problem 6 Geometry

The is year's IMO problem 6 was a geometry problem that only 6 participants managed to solve completely. The problem is formulated like this:

Let $$ABC$$ be an acute triangle with circumcircle $$\Gamma$$. Let $$\ell$$ be a tangent line to $$\Gamma$$, and let $$\ell_a$$, $$\ell_b$$ and $$\ell_c$$ be the lines obtained by reflecting $$\ell$$ in the lines $$BC$$, $$CA$$ and $$AB$$, respectively

Show that the circumcircle of the triangle determined by the lines $$\ell_a$$, $$\ell_b$$ and $$\ell_c$$ is tangent to the circle $$\Gamma$$.

A few solutions were found to this problem: using inversions, complex numbers, angle chasing, etc. My question is if we can reduce the problem to a simpler one in the following way:

Can we construct a triangle $$\Delta$$ for which $$\Gamma$$ is the incircle and $$\Gamma_1$$ is the 9 point circle? Of course, the answer should be yes if the circles are tangent and the radius of $$\Gamma_1$$ is greater than the radius of $$\Gamma$$. In this way we just apply a well known theorem of Feuerbach which says that the incircle and 9 point circle are tangent. How could we construct the triangle $$\Delta$$, starting from $$ABC$$?

This was my first idea when I saw the problem but didn't manage to finalize it.

• Hmmm... I had a talk with my younger brother today, and he said that the $\mathsf{second \: problem}$ was the toughest and very few managed to solve it completely.
– user9413
Commented Jul 23, 2011 at 20:06
• @chandru: Take a look at the results and see for yourself: imo-official.org/… . There are way more people who solved problem 2. Commented Aug 5, 2011 at 11:59
• Dear Beni, I am not arguing with you. I just pointed out that, the 2nd problem was also quite tough. That's all. Lets not make a big issue of it .
– user9413
Commented Aug 5, 2011 at 12:56

This link may be useful.There are quite a few solutions there.

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2365045&sid=0cdd97cc9547c2079a4ba23c56ba8f74#p2365045

In fact,this was the toughest problem at the IMO 2011.It was G8 on the Shortlist,meaning a hard problem. The IMO committee actually ended up misjudging the difficulty of the problems,as evident from the way they were numbered on the Shortlist.

• I don't see an answer to the question there. Did you have some special answer in mind, or did you just link to a list of solutions? Also, could you explain what G8 means? To me this is the group of eight :)
– t.b.
Commented Jul 31, 2011 at 8:04
• If you look at post number 18 in the link, you'll find a related idea, but there is no proof to it. Commented Aug 5, 2011 at 9:08
• I really apologise for not being clear. IMO problems are shortlisted as Algebra(A),Geometry(G),Combinatorics(C) and Number Theory(NT). There are 6-8 problems per topic.So, a Geometry problem may be designated as G1,G2..,G8(depending on how many problems are shortlisted).In this case ,problem 6 was designated as G8(Geometry 8) meaning a super hard problem(As the number increases, so does the level of difficulty) Commented Aug 26, 2011 at 15:22

Solution of problem 6 IMO 2011: I use the method of analytic geometry. Starting with the unit circle and 3 arbitrary points A,B C on its circumference, I found after laborious computations the equation of the second circumscribed circle. It is possible to construct the equation which is the tangent-condition of the 2 circles. Substitution completes the proof. In the case of a isosceles triangle the centre M of the second circle describes a limacon of Pascal, which degenerates in a circle in the case of a equilateral triangle Then that circle coincides the original unit circle