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here I am giving you question paper for International Mathematical Olympiad Examination in PDF file attached with it so you can get it easily.. some questions are given below : Geometry G1. In the triangle ABC the point J is the center of the excircle opposite to A. This excircle is tangent to the side BC at M, and to the lines AB and AC at K and L respectively. The lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of intersection of the lines AF and BC, and let T be the point of intersection of the lines AG and BC. Prove that M is the midpoint of ST. G2. Let ABCD be a cyclic quadrilateral whose diagonals AC and BD meet at E. The extensions of the sides AD and BC beyond A and B meet at F. Let G be the point such that ECGD is a parallelogram, and let H be the image of E under reflection in AD. Prove that D, H, F, G are concyclic. G3. In an acute triangle ABC the points D, E and F are the feet of the altitudes through A, B and C respectively. The incenters of the triangles AEF and BDF are I1 and I2 respectively; the circumcenters of the triangles ACI1 and BCI2 are O1 and O2 respectively. Prove that I1I2 and O1O2 are parallel. G4. Let ABC be a triangle with AB 6= AC and circumcenter O. The bisector of ∠BAC intersects BC at D. Let E be the reflection of D with respect to the midpoint of BC. The lines through D and E perpendicular to BC intersect the lines AO and AD at X and Y respectively. Prove that the quadrilateral BXCY is cyclic. G5. Let ABC be a triangle with ∠BCA = 90◦, and let C0 be the foot of the altitude from C. Choose a point X in the interior of the segment CC0, and let K, L be the points on the segments AX,BX for which BK = BC and AL = AC respectively. Denote by M the intersection of AL and BK. Show that MK = ML. G6. Let ABC be a triangle with circumcenter O and incenter I. The points D, E and F on the sides BC, CA and AB respectively are such that BD + BF = CA and CD + CE = AB. The circumcircles of the triangles BFD and CDE intersect at P 6= D. Prove that OP = OI. G7. Let ABCD be a convex quadrilateral with non-parallel sides BC and AD. Assume that there is a point E on the side BC such that the quadrilaterals ABED and AECD are circumscribed. Prove that there is a point F on the side AD such that the quadrilaterals ABCF and BCDF are circumscribed if and only if AB is parallel to CD. G8. Let ABC be a triangle with circumcircle ω and ℓ a line without common points with ω. Denote by P the foot of the perpendicular from the center of ω to ℓ. The side-lines BC,CA,AB intersect ℓ at the points X, Y, Z different from P. Prove that the circumcircles of the triangles AXP,BY P and CZP have a common point different from P or are mutually tangent at P. International Mathematical Olympiad Exam Question Papers Last edited by Sashwat; July 1st, 2019 at 04:45 PM. |