Olympiad problems

1988 All Soviet Union Mathematical Olympiad, 467

Tags: ratio , fixed , geometry , cyclic , midpoint

The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

2020 Czech-Austrian-Polish-Slovak Match, 6

Tags: equal angles , fixed , fixed point , geometry

Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point. (Dominik Burek, Poland)

2016 Vietnam Team Selection Test, 3

Tags: fixed point , fixed , circles , circumcircle , geometry

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$, $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$. The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ($G \ne A$). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$. Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$. On $EF$ take $Q$ for $QP = QI$. Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$. Prove that the midpoint of $IJ$ belongs to a fixed circle.

2022 Bulgarian Autumn Math Competition, Problem 10.2

Tags: geometry , fixed

Fix a triangle $ABC$. The variable point $M$ in its interior is such that $\angle MAC = \angle MBC$ and $N$ is the reflection of $M$ with respect to the midpoint of $AB$. Prove that $|AM| \cdot |BM| + |CM| \cdot |CN|$ is independent of the choice of $M$.

2020 Dutch IMO TST, 2

Tags: reflection , fixed point , fixed , circumcircle , angle bisector , geometry

Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.

VMEO IV 2015, 11.2

Tags: projection , geometry , fixed point , fixed , isogonal

Let $ABC$ be a triangle with two isogonal points $ P$ and $Q$ . Let $D, E$ be the projection of $P$ on $AB$, $AC$. $G$ is the projection of $Q$ on $BC$. $U$ is the projection of $G$ on $DE$, $ L$ is the projection of $P$ on $AQ$, $K$ is the symmetric of $L$ wrt $UG$. Prove that $UK$ passes through a fixed point when $P$ and $Q$ vary.

2016 Dutch IMO TST, 4

Tags: geometry , circles , fixed

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4

Tags: fixed , geometry , fixed point , length , circumcircle , angle

The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that: a) the segment $MN$ has a constant length, b) all circles circumscribed around triangle $DMN$ have a common point

1991 Tournament Of Towns, (295) 2

Tags: geometry , fixed , fixed point

The chord $MN$ on the circle is fixed. For every diameter $AB$ of the circle consider the intersection point $C$ of the lines $AM$ and $BN$ and construct the line $\ell$ passing through $C$ perpendicularly to $AB$. Prove that all the lines $\ell$ pass through a fixed point. (E. Kulanin, Moscow)

1924 Eotvos Mathematical Competition, 2

Tags: geometry , locus , fixed

If $O$ is a given point, $\ell$ a given line, and $a$ a given positive number, find the locus of points $P$ for which the sum of the distances from $P$ to $O$ and from $P$ to $\ell$ is $a$.

2016 Regional Olympiad of Mexico West, 3

Tags: geometry , fixed

A circle $\omega$ with center $O$ and radius $r$ is constructed. A point $P$ is chosen on the circumference $\omega$ and a point A is taken inside it, such that is outside the line that passes through $P$ and $O$. Point $B$ is constructed, the reflection of $A$ wrt $O$. and $P'$ is another point on the circumference such that the chord $PP'$ is perpendicular to $PA$. Let $Q$ be the point on the line $PP'$ that minimizes the sum of distances from $A$ to $Q$ and from $Q$ to $B$. Show that the value of the sum of the lengths $AQ+QB$ does not depend on the choice of points $P$ or $A$

2009 China Northern MO, 6

Tags: geometry , fixed , area

Given a minor sector $AOB$ (Here minor means that $ \angle AOB <90$). $O$ is the centre , chose a point $C$ on arc $AB$ ,Let $P$ be a point on segment $OC$ , join $AP$ , $BP$ , draw a line through $B$ parallel to $AP$ , the line meet $OC$ at point $Q$ ,join $AQ$ . Prove that the area of polygon $AQPBO$ does not change when points $P,C$ move . [img]https://cdn.artofproblemsolving.com/attachments/3/e/4bdd3a20fe1df3fce0719463b55ef93e8b5d7b.png[/img]

2006 Sharygin Geometry Olympiad, 22

Tags: geometry , fixed , circumcenter , locus

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.

2016 Dutch IMO TST, 4

Tags: geometry , circles , fixed

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.