MathDB

3

Part of 2009 Serbia Team Selection Test

Problems(2)

Biggest number of sets

Source: Serbia TST 2009

4/17/2009
Find the largest natural number n n for which there exist different sets S1,S2,…,Sn S_1,S_2,\ldots,S_n such that: 1∘ 1^\circ ∣Si∪Sj∣≤2004 |S_i\cup S_j|\leq 2004 for each two 1≤i,j≤n 1\leq i,j\le n and 2∘ 2^\circ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\} for each three integers 1≤i<j<k≤n 1\le i<j<k\le n.
floor functioncombinatorics unsolvedcombinatorics
Circles and collinearity

Source: Serbia TST 2009

4/17/2009
Let k k be the inscribed circle of non-isosceles triangle △ABC \triangle ABC, which center is S S. Circle k k touches sides BC,CA,AB BC,CA,AB in points P,Q,R P,Q,R respectively. Line QR QR intersects BC BC in point M M. Let a circle which contains points B B and C C touch k k in point N N. Circumscribed circle of △MNP \triangle MNP intersects line AP AP in point L L, different from P P. Prove that points S,L S,L and M M are collinear.
geometrycircumcircleratiogeometric transformationhomothetyexterior angleangle bisector