MathDB

3

Part of 2011 Serbia National Math Olympiad

Problems(2)

collinear iff X is circumcenter Serbia Math Olyimpiad 2011

Source:

4/8/2011
Let HH be orthocenter and OO circumcenter of an acuted angled triangle ABCABC. DD and EE are feets of perpendiculars from AA and BB on BCBC and ACAC respectively. Let ODOD and OEOE intersect BEBE and ADAD in KK and LL, respectively. Let XX be intersection of circumcircles of HKDHKD and HLEHLE different than HH, and MM is midpoint of ABAB. Prove that K,L,MK, L, M are collinear iff XX is circumcenter of EODEOD.
geometrycircumcirclegeometry unsolved
T with 66 points

Source:

4/8/2011
Set TT consists of 6666 points in plane, and PP consists of 1616 lines in plane. Pair (A,l)(A,l) is good if A∈TA \in T, l∈Pl \in P and A∈lA \in l. Prove that maximum number of good pairs is no greater than 159159, and prove that there exits configuration with exactly 159159 good pairs.
linear algebramatrixfunctioninequalitiescombinatorics