Olympiad problems

1992 All Soviet Union Mathematical Olympiad, 575

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.

VI Soros Olympiad 1999 - 2000 (Russia), 10.10

Tags: geometry , fixed , fixed point

Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.

Kyiv City MO 1984-93 - geometry, 1984.8.1

Tags: geometry , area , fixed , symmetric

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of this quadrilateral does not depend on the choice of the point $M$.

2017 Hanoi Open Mathematics Competitions, 12

Tags: geometry , fixed , segment

Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?

Geometry Mathley 2011-12, 12.1

Tags: orthocenter , equal ratio , fixed , ratio

Let $ABC$ be an acute triangle with orthocenter $H$, and $P$ a point interior to the triangle. Points $D,E,F$ are the reflections of $P$ about $BC,CA,AB$. If $Q$ is the intersection of $HD$ and $EF$, prove that the ratio $HQ/HD$ is independent of the choice of $P$. Luis González

1957 Moscow Mathematical Olympiad, 367

Tags: area , geometry , fixed , square

Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.

1992 Austrian-Polish Competition, 5

Tags: geometry , circles , fixed point , fixed , product

Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.

Mathley 2014-15, 1

Tags: circles , geometry , fixed

Let $ABC$ be an acute triangle inscribed in a circle $(O)$ that is fixed, and two of the vertices $B$, $C$ are fixed while vertex $A$ varies on the circumference of the circle. Let $I$ be the center of the incircle, and $AD$ the angle bisector. Let $K$, $L$ be the circumcenters of $CAD$, $ABD$. A line through $O$ parallel to $DL$, $DK$ intersects the line that is through $I$ perpendicular to $IB$, $IC$ at $M$, $N$ respectively. Prove that $MN$ is tangent to a fixed circle when $A$ varies on the circle $(O)$. Tran Quang Hung, Natural Science High School, National University, Hanoi

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 $M$.

2012 Oral Moscow Geometry Olympiad, 5

Tags: geometry , circumcircle , orthocenter , fixed , fixed point

Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of altitudes of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.

VMEO IV 2015, 11.2

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

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.

2012 Czech-Polish-Slovak Junior Match, 3

Tags: geometry , circumcircle , projection , equilateral , fixed , area of a triangle

Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

1990 Greece National Olympiad, 2

Tags: geometry , vector , fixed

Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$ is independent of positions of $K,L,M,N$, (i.e. is constant )

1959 Polish MO Finals, 2

Tags: geometry , fixed , equilateral

In an equilateral triangle $ ABC $, point $ O $ is chosen and perpendiculars $ OM $, $ ON $, $ OP $ are dropped to the sides $ BC $, $ CA $, $ AB $, respectively. Prove that the sum of the segments $ AP $, $ BM $, $ CN $ does not depend on the position of point $ O $.