Geometry Shortlist Problem in IMO 2015

Problem 1.

Let ABC be an acute triangle with orthocenter H. Let G be the point such that the quadrilateral ABGH is a parallelogram. Let I be the point on the line GH such that AC bisects HI. Suppose that the line AC intersects the circumcircle of the triangle GCI at C and J. Prove that IJ = AH.

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Problem 2.

Let ABC be a triangle inscribed into a circle \Omega with center O. A circle \Gamma with center A meets the side BC at points D and E such that D lies between B and E. Moreover, let F and G be the common points of \Gamma and \Omega. We assume that F lies on the arc AB of \Omega not containing C, and G lies on the arc AC of \Omega not containing B. The circumcircles of the triangles BDF and CEG meet the sides AB and AC again at K and L, respectively. Suppose that the lines FK and GL are distinct and intersect at X. Prove that the points A, X, and O are collinear.

Solution Problem 2

Problem 3.

Let ABC be a triangle with \angle C=90^0, and let H be the foot of the altitude from C. A point D is chosen inside the triangle CBH so that CH bisects AD. Let P be the intersection point of the lines BD and CH. Let \omega be the semicircle with diameter BD that meets the segment CB at an interior point. A line through P is tangent to \omega at Q. Prove that the lines CQ and AD meet on \omega.

Problem 4.

Let ABC be an acute triangle, and let M be the midpoint of AC. A circle \omega passing through B and M meets the sides AB and BC again at P and Q, respectively. Let T be the point such that the quadrilateral BPTQ is a parallelogram. Suppose that T lies on the circumcircle of the triangle ABC. Determine all possible values of BT / BM.

Problem 5.

Let ABC be a triangle with CA\neq CB. Let D, F, and G be the midpoints of the sides AB, AC, and BC, respectively. A circle G passing through C and tangent to AB at D meets the segments AF and BG at H and I, respectively. The points H’ and I’ are symmetric to H and I about F and G, respectively. The line H’I’ meets CD and FG at Q and M, respectively. The line CM meets \Gamma again at P. Prove that CQ = QP.

Problem 6.

Let ABC be an acute triangle with AB > AC, and let G be its circumcircle. Let H, M, and F be the orthocenter of the triangle, the midpoint of BC, and the foot of the altitude from A, respectively. Let Q and K be the two points on \Gamma that satisfy \angle AQH=90^0 and \angle QKH=90^0. Prove that the circumcircles of the triangles KQH and KFM are tangent to each other.

Problem 7.

Let ABCD be a convex quadrilateral, and let P, Q, R, and S be points on the sides AB, BC, CD, and DA, respectively. Let the line segments PR and QS meet at O. Suppose that each of the quadrilaterals APOS, BQOP, CROQ, and DSOR has an incircle. Prove that the lines AC, PQ, and RS are either concurrent or parallel to each other.

Problem 8.

A triangulation of a convex polygon \Pi is a partitioning of \Pi into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a Thai angulation if all triangles in it have the same area. Prove that any two different Thai angulations of a convex polygon \Pi differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thai angulation with a different pair of triangles so as to obtain the second Thai angulation.)

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