Olympiad problems

1990 Greece National Olympiad, 2

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 )

1949-56 Chisinau City MO, 62

Tags: geometry , 3d geometry , tetrahedron , volume , fixed , constant

On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.

IV Soros Olympiad 1997 - 98 (Russia), 9.11

Tags: geometry , fixed , fixed point

Given two circles intersecting at points $A$ and $B$. A certain circle touches the first at point $A$, intersects the second at point $M$ and intersects the straight line $AB$ at point $P$ ($M$ and $P$ are different from $B$). Prove that the straight line $MP$ passes through a fixed point of the plane (for any change in the third circle).

2014 Thailand TSTST, 1

Tags: geometry , fixed , fixed angle

In a triangle $ABC, AC = BC$ and $D$ is the midpoint of $AB$. Let $E$ be an arbitrary point on line $AB$ which is not $B$ or $D$. Let $O$ be the circumcenter of $\vartriangle ACE$ and $F$ the intersection of the perpendicular from $E$ to $BC$ and the perpendicular to $DO$ at $D$. Prove that the acute angle between $BC$ and $BF$ does not depend on the choice of point $E$.

1911 Eotvos Mathematical Competition, 2

Tags: geometry , fixed , constant , octagon

Let $Q$ be any point on a circle and let $P_1P_2P_3...P_8$ be a regular inscribed octagon. Prove that the sum of the fourth powers of the distances from $Q$ to the diameters $P_1P_5$, $P_2P_6$, $P_3P_7$, $P_4P_8$ is independent of the position of $Q$.

Ukraine Correspondence MO - geometry, 2006.10

Tags: radii , geometry , mixtilinear excircle , fixed , isosceles

Let $ABC$ be an isosceles triangle ($AB=AC$). An arbitrary point $M$ is chosen on the extension of the $BC$ beyond point $B$. Prove that the sum of the radius of the circle inscribed in the triangle $AMB$ and the radius of the circle tangent to the side $AC$ and the extensions of the sides $AM, CM$ of the triangle $AMC$ does not depend on the choice of point $M$.

2012 Czech-Polish-Slovak Junior Match, 3

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

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 $BC$

2017 Peru Iberoamerican Team Selection Test, P1

Tags: ratio , geometry , perimeter , fixed , constant , tangent circle

Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.

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 $.

2022 Bulgarian Autumn Math Competition, Problem 10.2

Tags: fixed , geometry

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 $M$.

2007 Oral Moscow Geometry Olympiad, 6

Tags: geometry , fixed , fixed point , tangent

A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point. (A. Zaslavsky)

2018 Thailand TSTST, 3

Tags: geometry , fixed , perpendicular , fixed point

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

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 $A$

1992 All Soviet Union Mathematical Olympiad, 575

Tags: 3d geometry , geometry , plane , fixed , product , sphere

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 $p$.

1993 Tournament Of Towns, (395) 3

Tags: geometry , fixed , hexagon , equilateral

Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined. (V Proizvolov)

2021 Mediterranean Mathematics Olympiad, 3

Tags: geometry , incenter , equilateral , fixed point , fixed

Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$. (The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)

2007 Oral Moscow Geometry Olympiad, 6

Tags: geometry , fixed , tangent

A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle. (A. Zaslavsky)

2022 Adygea Teachers' Geometry Olympiad, 2

Tags: geometry , trapezoid , fixed , fixed point

An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.

Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.3

Tags: geometry , incenter , fixed , fixed point , equilateral

The equilateral triangle $ABC$ is inscribed in the circle $w$. Points $F$ and $E$ on the sides $AB$ and $AC$, respectively, are chosen such that $\angle ABE+ \angle ACF = 60^o$. The circumscribed circle of $\vartriangle AFE$ intersects the circle $w$ at the point $D$ for the second time. The rays $DE$ and $DF$ intersect the line $BC$ at the points $X$ and $Y$, respectively. Prove that the center of the inscribed circle of $\vartriangle DXY$ does not depend on the choice of points $F$ and $E$. (Hilko Danilo)

1966 Poland - Second Round, 6

Tags: geometry , fixed , equilateral

Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line $ p $ of the plane of this triangle does not depend on the position of the line $ p $ if and only if it the triangle is equilateral.

2017 Ukrainian Geometry Olympiad, 3

Tags: geometry , fixed , intersect , circles , locus , locus problems

Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.

2010 Oral Moscow Geometry Olympiad, 6

Tags: geometry , perpendicular bisector , circumcircle , fixed , circles

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

Swiss NMO - geometry, 2022.8

Tags: incenter , fixed , geometry

Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.

Kyiv City MO Seniors 2003+ geometry, 2007.11.5

Tags: circumcircle , equal angles , fixed point , fixed , geometry

The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$. (Alexei Klurman)

1989 Romania Team Selection Test, 4

Tags: sphere , fixed , tangent spheres , 3d geometry , trihedral angle , geometry

Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively. Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$. Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.