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Germany 2015-21 (De_MO) 12p

geometry problems from German National Mathematical Olympiads
with aops links in the names

2015 - 2021

2015 German MO P5
Let ABCD be a convex quadrilateral such that the circle with diameter AB touches the line CD. Prove that that the circle with diameter CD touches the line AB if and only if BC and AD are parallel.

2016 German MO P3
Let I_a be the A-excenter of a scalene triangle ABC. And let M be the point symmetric to I_a about line BC.Prove that line AM is parallel to the line through the circumcenter and the orthocenter of triangle I_aCB.

2016 German MO P5
Let A,B,C,D be points on a circle with radius r in this order such that |AB|=|BC|=|CD|=s and |AD|=s+r. Find all possible values of the interior angles of the quadrilateral ABCD.

2017 German MO P2
Let ABC be a triangle such that \vert AB\vert \ne \vert AC\vert. Prove that there exists a point D \ne A on its circumcircle satisfying the following property:
For any points M, N outside the circumcircle on the rays AB and AC, respectively, satisfying \vert BM\vert=\vert CN\vert, the circumcircle of AMN passes through D

2017 German MO P4
Let ABCD be a cyclic quadrilateral. The point P is chosen on the line AB such that the circle passing through C,D and P touches the line AB. Similarly, the point Q is chosen on the line CD such that the circle passing through A,B and Q touches the line CD.
Prove that the distance between P and the line CD equals the distance between Q and AB.

2018 German MO P3
We are given a tetrahedron with two edges of length a and the remaining four edges of length b where a and b are positive real numbers. What is the range of possible values for the ratio v=a/b?

2018 German MO P6
Let P be a point in the interior of a triangle ABC and let the rays \overrightarrow{AP}, \overrightarrow{BP} and \overrightarrow{CP} intersect the sides BC, CA and AB in A_1,B_1 and C_1, respectively. Let D be the foot of the perpendicular from A_1 to B_1C_1. Show that
\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.

2019 German MO P2
Let a and b be two circles, intersecting in two distinct points Y and Z. A circle k touches the circles a and b externally in the points A and B.
Show that the angular bisectors of the angles \angle ZAY and \angle YBZ intersect on the line YZ.

2020 German MO P1
Let k be a circle with center M and let B be another point in the interior of k. Determine those points V on k for which \measuredangle BVM becomes maximal.

2020 German MO P6
The insphere and the exsphere opposite to the vertex D of a (not necessarily regular) tetrahedron ABCD touch the face ABC in the points X and Y, respectively. Show that \measuredangle XAB=\measuredangle CAY.

Let P on AB, Q on BC, R on CD and S on AD be points on the sides of a convex quadrilateral ABCD. Show that the following are equivalent:
(1) There is a choice of P,Q,R,S, for which all of them are interior points of their side, such that PQRS has minimal perimeter.
(2) ABCD is a cyclic quadrilateral with circumcenter in its interior.

Let OFT and NOT be two similar triangles (with the same orientation) and let FANO be a parallelogram. Show that \vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.


sources:
https://www.mathe-wettbewerbe.de/mo/
https://www.mathematik-olympiaden.de/moev/index.php/aufgaben/aufgabenarchiv

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