MathDB

2

Part of 2007 Italy TST

Problems(2)

Cyclical 3-subsets

Source: Italian TST , day 1, n°2

6/1/2007
In a competition, there were 2n+12n+1 teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team A,B,C{ A,B,C } if AA won against BB, BB won against CC , CC won against AA. (a) Find the minimum of cyclical 3-subset (depending on nn); (b) Find the maximum of cyclical 3-subset (depending on nn).
modular arithmeticcombinatorics proposedcombinatorics
Triangle of circumcenters...

Source: Italian TST , day 2, n°2

6/2/2007
Let ABCABC a acute triangle. (a) Find the locus of all the points PP such that, calling Oa,Ob,OcO_{a}, O_{b}, O_{c} the circumcenters of PBCPBC, PACPAC, PABPAB: OaObAB=ObOcBC=OcOaCA\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA} (b) For all points PP of the locus in (a), show that the lines AOaAO_{a}, BObBO_{b} , COcCO_{c} are cuncurrent (in XX); (c) Show that the power of XX wrt the circumcircle of ABCABC is: −a2+b2+c2−5R24-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4 Where a=BCa=BC , b=ACb=AC and c=ABc=AB.
geometrycircumcirclegeometry proposed