Let be the circumcircle of right-angled triangle ABC (A = 90).
Proposed by Nikolai Beluhov (Bulgaria) and Morteza SaghafianĤ.
Find all positive integers N such that there exists a triangle which canbe dissected into N similar quadrilaterals. Show that AD is bisector of the angle CAP. The tangent to C2 from Q intersects C1and C2 in C and D, respectively (The points A and D lie on different sides ofthe line PQ). The tangent toC1 at A intersects C2 in P and the line PB intersects C1 for the second timein Q (suppose that Q is outside C2). Let two circles C1 and C2 intersect in points A and B. Show that the length of segment XY is not more thanhalf of the perimeter of ABCD.Ģ. Let X and Y be two arbitrary points on 1and 2, respectively. In trapezoid ABCD with AB CD, 1 and 2 are two circles with di-ameters AD and BC, respectively. Problems of 3rd Iranian Geometry Olympiad 2016 (Medium)ġ. Let ABCD be a convex quadrilateral with these properties:ADC = 135 and ADBABD = 2DAB = 4CBD. If the line CL be the internal bisector of angle C, findall possible values for angles B and C.ĥ. In a right-angled triangle ABC (A = 90), the perpendicular bisectorof BC intersects the line AC in K and the perpendicular bisector of BK inter-sects the line AB in L. Show that among these 4 newpoints, there is only one point inside the quadrilateral ABCD.Ĥ. Makea parallelogram on each two consecutive sides. Suppose that ABCD is a convex quadrilateral with no parallel sides. The line XY inter-sects for the second time in point P. Let X be apoint on AC and Y be a point on the circle, such that CX = CY = AB.(The points A and Y lie on different sides of the line BC). Let be the circumcircle of triangle ABC with AC > AB. Only draw the path and write its length.Ģ. Help Ali to find the shortest pathbetween A and B.
Geometry olympiad problems pdf free#
He cannot walk inside theblack areas but he is free to move in any direction inside the white areas (notonly the grid lines but the whole plane). Ali wants to move from point A to point B. Problems of 3rd Iranian Geometry Olympiad 2016 (Elementary)ġ.
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