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3

Part of 2011 Middle European Mathematical Olympiad

Problems(2)

Ah, don't bother me with this much point (show AE prp KL)

Source: Middle European Mathematical Olympiad 2011 - Individuals I-3

9/6/2011
In a plane the circles K1\mathcal K_1 and K2\mathcal K_2 with centers I1I_1 and I2I_2, respectively, intersect in two points AA and BB. Assume that ∠I1AI2\angle I_1AI_2 is obtuse. The tangent to K1\mathcal K_1 in AA intersects K2\mathcal K_2 again in CC and the tangent to K2\mathcal K_2 in AA intersects K1\mathcal K_1 again in DD. Let K3\mathcal K_3 be the circumcircle of the triangle BCDBCD. Let EE be the midpoint of that arc CDCD of K3\mathcal K_3 that contains BB. The lines ACAC and ADAD intersect K3\mathcal K_3 again in KK and LL, respectively. Prove that the line AEAE is perpendicular to KLKL.
geometrycircumcirclegeometric transformationreflection
max. of points with no triangles

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-3

9/6/2011
For an integer n≥3n \geq 3, let M\mathcal M be the set {(x,y)∣x,y∈Z,1≤x≤n,1≤y≤n}\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\} of points in the plane.
What is the maximum possible number of points in a subset S⊆MS \subseteq \mathcal M which does not contain three distinct points being the vertices of a right triangle?
combinatorics unsolvedcombinatorics