geometry problems from German National Mathematical Olympiads
with aops links in the names
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 $\measuredangle XAB=\measuredangle CAY$.
sources:
https://www.mathe-wettbewerbe.de/mo/
https://www.mathematik-olympiaden.de/moev/index.php/aufgaben/aufgabenarchiv
with aops links in the names
2015 - 2021
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 $\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.$
https://www.mathe-wettbewerbe.de/mo/
https://www.mathematik-olympiaden.de/moev/index.php/aufgaben/aufgabenarchiv
No comments:
Post a Comment