Olympiad problems

2005 Canada National Olympiad, 3

Let $S$ be a set of $n\ge 3$ points in the interior of a circle. $a)$ Show that there are three distinct points $a,b,c\in S$ and three distinct points $A,B,C$ on the circle such that $a$ is (strictly) closer to $A$ than any other point in $S$, $b$ is closer to $B$ than any other point in $S$ and $c$ is closer to $C$ than any other point in $S$. $b)$ Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.

2012 Serbia Team Selection Test, 3

Tags: radical axis , circumcircle , geometry

Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$. a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic. b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.

2009 Ukraine National Mathematical Olympiad, 3

Tags: geometric transformation , parallelogram , circumcircle , geometry , cyclic quadrilateral

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

2019 Philippine MO, 4

Tags: geometry , concyclic , circumcircle , equal angles

In acute triangle $ABC $with $\angle BAC > \angle BCA$, let $P$ be the point on side $BC$ such that $\angle PAB = \angle BCA$. The circumcircle of triangle $AP B$ meets side $AC$ again at $Q$. Point $D$ lies on segment $AP$ such that $\angle QDC = \angle CAP$. Point $E$ lies on line $BD$ such that $CE = CD$. The circumcircle of triangle $CQE$ meets segment $CD$ again at $F$, and line $QF$ meets side $BC$ at $G$. Show that $B, D, F$, and $G$ are concyclic

2011 Sharygin Geometry Olympiad, 8

Tags: convex polygon , diagonal , geometry , circumcircle

A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?

2001 Czech-Polish-Slovak Match, 2

Tags: geometric transformation , perimeter , parallelogram , reflection , circumcircle , geometry

A triangle $ABC$ has acute angles at $A$ and $B$. Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$. Let $S$ be the circumcenter of $\triangle ABC$. Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.

2019 Bulgaria National Olympiad, 2

Tags: circumcenter , orthocenter , geometry , perpendicular bisector , circumcircle

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$

2014 Contests, 3

Tags: circumcircle , geometry , combinatorics

Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices. a) Find all possible values of pair $(A,B).$ b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

2005 Mexico National Olympiad, 1

Tags: circumradius , geometry , circumcircle

Let $O$ be the center of the circumcircle of an acute triangle $ABC$, let $P$ be any point inside the segment $BC$. Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$. (i) Consider the triangle $PQR$, show that it is similar to triangle $ABC$ and that $O$ is its orthocenter. (ii) Show that the circumcircles of triangles $BPO$, $COP$, $PQR$ have the same radius.

2022 Thailand TST, 2

Tags: projective geometry , geometry , circumcircle

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

2012 Sharygin Geometry Olympiad, 21

Tags: rotation , geometric transformation , circumcircle , geometry

Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments. [i]Nikolai Beluhov and Emil Kolev[/i]

2022 Bosnia and Herzegovina IMO TST, 1

Tags: reflection , geometry , geometric transformation , circumcircle

Let $ABC$ be a triangle such that $AB=AC$ and $\angle BAC$ is obtuse. Point $O$ is the circumcenter of triangle $ABC$, and $M$ is the reflection of $A$ in $BC$. Let $D$ be an arbitrary point on line $BC$, such that $B$ is in between $D$ and $C$. Line $DM$ cuts the circumcircle of $ABC$ in $E,F$. Circumcircles of triangles $ADE$ and $ADF$ cut $BC$ in $P,Q$ respectively. Prove that $DA$ is tangent to the circumcircle of triangle $OPQ$.

2024 Moldova EGMO TST, 5

Tags: geometric inequality , geometry , angle bisector , circumcircle , inequalities

$AD$ Is the angle bisector Of $\angle BAC$ Where $D$ lies on the The circumcircle of $\triangle ABC$. Show that $2AD>AB+AC$

1987 IMO Longlists, 22

Tags: geometry , triangle , minimization , circumcenter , circumcircle

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]

1998 IMO Shortlist, 2

Tags: geometry , circumcircle , ratio , area , trapezoid

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

2005 Postal Coaching, 10

Tags: trig identities , law of sines , geometry , trigonometry , circumcircle , cyclic quadrilateral

On the sides $AB$ and $BC$ of triangle $ABC$, points $K$ and $M$ are chosen such that the quadrilaterals $AKMC$ and $KBMN$ are cyclic , where $N = AM \cap CK$ . If these quads have the same circumradii, find $\angle ABC$

2005 MOP Homework, 2

Tags: incenter , geometry , search , circumcircle

Let $I$ be the incenter of triangle $ABC$, and let $A_1$, $B_1$, and $C_1$ be arbitrary points lying on segments $AI$,$BI$, and $CI$, respectively. The perpendicular bisectors of segments $AA_1$, $BB_1$, and $CC_1$ form triangles $A_2B_2C_2$. Prove that the circumcenter of triangle $A_2B_2C_2$ coincides with the circumcenter of triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$.

2010 Middle European Mathematical Olympiad, 10

Tags: ratio , parallelogram , circumcircle , geometry , cyclic quadrilateral

Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\sphericalangle{AFS}=\sphericalangle{ECD}$. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 6)[/i]

2004 Germany Team Selection Test, 3

Tags: trigonometry , conic , analytic geometry , incenter , geometry , ratio , circumcircle

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

2003 China National Olympiad, 1

Tags: incenter , circumcircle , geometry

Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$. [i]Shen Wunxuan[/i]

2014 Germany Team Selection Test, 2

Tags: incenter , circumcircle , geometry , cyclic quadrilateral

Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$. Prove \[ AN \cdot NC = CD \cdot BN. \]

2023 Romanian Master of Mathematics Shortlist, G1

Tags: concurrency , angle chasing , circumcircle , trigonometry , geometry

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. The incircle of the triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. The circumcircle of triangle $ADI$ crosses $\omega$ again at $P$, and the lines $PE$ and $PF$ cross $\omega$ again at $X$and $Y$, respectively. Prove that the lines $AI$, $BX$ and $CY$ are concurrent.

2014 Bosnia Herzegovina Team Selection Test, 3

Tags: perpendicular bisector , angle bisector , circumcircle , geometry

Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.

2015 South Africa National Olympiad, 4

Tags: geometry , circumcircle

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.

1992 IberoAmerican, 2

Tags: trigonometry , rhombus , circumcircle , trapezoid , geometry , perpendicular bisector

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.