International Mathematical Olympiad Exam Previous Question Papers

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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

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