Aops Community Instructive Olympiad Geometry Problems: at A National Olympiad and Higher Level

AoPS Community Instructive Olympiad Geometry Problems
At a national Olympiad and higher level.

www.artofproblemsolving.com/community/c53840
by v Enhance, Davi Medeiros, leader, aZpElr68Cb51U51qy9OM, syk0526, WakeUp, stef mol, Goutham,
Kimpul, brianchung11, jgnr, mathpk, MathPanda1, Arne, grobber, shobber, djb86, tipe, yetti, aditya21, Wol-
stenholme
APMO 2014 Q5 Circles ω and Ω meet at points A and B. Let M be the midpoint of the arc AB of circle
ω (M lies inside Ω). A chord M P of circle ω intersects Ω at Q (Q lies inside ω). Let `P be the
tangent line to ω at P , and let `Q be the tangent line to Ω at Q. Prove that the circumcircle of
the triangle formed by the lines `P , `Q and AB is tangent to Ω.
Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan
APMO 2013 Q5 Let ABCD be a quadrilateral inscribed in a circle ω, and let P be a point on the ex-
tension of AC such that P B and P D are tangent to ω. The tangent at C intersects P D at Q
and the line AD at R. Let E be the second point of intersection between AQ and ω. Prove that
B, E, R are collinear.
APMO 2012 Q4 Let ABC be an acute triangle. Denote by D the foot of the perpendicular line drawn
from the point A to the side BC, by M the midpoint of BC, and by H the orthocenter of ABC.
Let E be the point of intersection of the circumcircle Γ of the triangle ABC and the half line
M H, and F be the point of intersection (other than E) of the line ED and the circle Γ. Prove
that BF
CF = AC must hold.
AB
(Here we denote XY the length of the line segment XY .)
APMO 2011 Q4 Let ABC be an acute triangle with ∠BAC = 30◦ . The internal and external angle
bisectors of ∠ABC meet the line AC at B1 and B2 , respectively, and the internal and external
angle bisectors of ∠ACB meet the line AB at C1 and C2 , respectively. Suppose that the circles
with diameters B1 B2 and C1 C2 meet inside the triangle ABC at point P . Prove that ∠BP C =
90◦ .
APMO 2010 Q4 Let ABC be an acute angled triangle satisfying the conditions AB > BC and AC >
BC. Denote by O and H the circumcentre and orthocentre, respectively, of the triangle ABC.
Suppose that the circumcircle of the triangle AHC intersects the line AB at M different from
A, and the circumcircle of the triangle AHB intersects the line AC at N different from A. Prove
that the circumcentre of the triangle M N H lies on the line OH.
APMO 2009 Q3 Let three circles Γ1 , Γ2 , Γ3 , which are non-overlapping and mutually external, be given
in the plane. For each point P in the plane, outside the three circles, construct six points
A1 , B1 , A2 , B2 , A3 , B3 as follows: For each i = 1, 2, 3, Ai , Bi are distinct points on the circle Γi
such that the lines P Ai and P Bi are both tangents to Γi . Call the point P exceptional if, from
© 2019 AoPS Incorporated 1

AoPS Community Instructive Olympiad Geometry Problems
the construction, three lines A1 B1 , A2 B2 , A3 B3 are concurrent. Show that every exceptional
point of the plane, if exists, lies on the same circle.
APMO 2008 Q3 Let Γ be the circumcircle of a triangle ABC. A circle passing through points A and
C meets the sides BC and BA at D and E, respectively. The lines AD and CE meet Γ again
at G and H, respectively. The tangent lines of Γ at A and C meet the line DE at L and M ,
respectively. Prove that the lines LH and M G meet at Γ.
APMO 2004 Q2 Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Prove
that the area of one of the triangles AOH, BOH and COH is equal to the sum of the areas of
the other two.
APMO 2000 Q3 Let ABC be a triangle. Let M and N be the points in which the median and the angle
bisector, respectively, at A meet the side BC. Let Q and P be the points in which the perpendic-
ular at N to N A meets M A and BA, respectively. And O the point in which the perpendicular
at P to BA meets AN produced.
Prove that QO is perpendicular to BC.
APMO 1998 Q4 Let ABC be a triangle and D the foot of the altitude from A. Let E and F lie on a
line passing through D such that AE is perpendicular to BE, AF is perpendicular to CF , and
E and F are different from D. Let M and N be the midpoints of the segments BC and EF ,
respectively. Prove that AN is perpendicular to N M .
APMO 1995 Q3 Let P QRS be a cyclic quadrilateral such that the segments P Q and RS are not par-
allel. Consider the set of circles through P and Q, and the set of circles through R and S.
Determine the set A of points of tangency of circles in these two sets.
APMO 1995 Q4 Let C be a circle with radius R and centre O, and S a fixed point in the interior of C.
Let AA0 and BB 0 be perpendicular chords through S. Consider the rectangles SAM B, SBN 0 A0 ,
SA0 M 0 B 0 , and SB 0 N A. Find the set of all points M , N 0 , M 0 , and N when A moves around the
whole circle.
EGMO 2015 Q6 Let H be the orthocentre and G be the centroid of acute-angled triangle ABC with
AB 6= AC. The line AG intersects the circumcircle of ABC at A and P . Let P 0 be the reflection
of P in the line BC. Prove that ∠CAB = 60 if and only if HG = GP 0
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AoPS Community Instructive Olympiad Geometry Problems

At a national Olympiad and higher level.

(Here we denote XY the length of the line segment XY .)

© 2019 AoPS Incorporated 1

Prove that QO is perpendicular to BC.

© 2019 AoPS Incorporated 2

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