Subcontests
(6)Three Element Subsets of {1,2,...,n}
For a given integer n≥3, let S1,S2,…,Sm be distinct three-element subsets of the set {1,2,…,n} such that for each 1≤i,j≤m;i=j the sets Si∩Sj contain exactly one element. Determine the maximal possible value of m for each n. Circles Tangent to Sides and Circumcircle
Let ABC be a triangle with circumcircle ω and let ωA be a circle drawn outside ABC and tangent to side BC at A1 and tangent to ω at A2. Let the circles ωB and ωC and the points B1,B2,C1,C2 are defined similarly. Prove that if the lines AA1,BB1,CC1 are concurrent, then the lines AA2,BB2,CC2 are also concurrent. Concyclicity Implies Perpendicularity
Let D be the midpoint of the side BC of a triangle ABC and AD intersect the circumcircle of ABC for the second time at E. Let P be the point symmetric to the point E with respect to the point D and Q be the point of intersection of the lines CP and AB. Prove that if A,C,D,Q are concyclic, then the lines BP and AC are perpendicular.