Olympiad problems

2014 CentroAmerican, 2

Tags: geometry , incenter , geometric transformation , reflection , trigonometry , circumcircle , angle bisector

Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that \[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]

2017 Iran MO (3rd round), 2

Tags: circumcircle , geometry

Assume that $P$ be an arbitrary point inside of triangle $ABC$. $BP$ and $CP$ intersects $AC$ and $AB$ in $E$ and $F$, respectively. $EF$ intersects the circumcircle of $ABC$ in $B'$ and $C'$ (Point $E$ is between of $F$ and $B'$). Suppose that $B'P$ and $C'P$ intersects $BC$ in $C''$ and $B''$ respectively. Prove that $B'B''$ and $C'C''$ intersect each other on the circumcircle of $ABC$.

2018 Saudi Arabia BMO TST, 1

Tags: radical axis , geometry , bisects segment

Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$. a) Prove that $DE$ is parallel to $BC$. b) Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.

2011 Mexico National Olympiad, 2

Tags: circumcircle , geometric transformation , reflection , power of a point , perpendicular bisector , geometry

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.

2016 Sharygin Geometry Olympiad, P22

Tags: geometry , 3d geometry , isogonal conjugate

Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$. Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C, $ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$. Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.

Kharkiv City MO Seniors - geometry, 2015.10.3

Tags: geometry , circumcircle , circles , equal segments

On side $AB$ of triangle $ABC$, point $M$ is selected. A straight line passing through $M$ intersects the segment $AC$ at point $N$ and the ray $CB$ at point $K$. The circumscribed circle of the triangle $AMN$ intersects $\omega$, the circumscribed circle of the triangle $ABC$, at points $A$ and $S$. Straight lines $SM$ and $SK$ intersect with $\omega$ for the second time at points $P$ and $Q$, respectively. Prove that $AC = PQ$.

2019 CHMMC (Fall), 4

Tags: geometry

Let $\vartriangle ABC$ be a triangle such that the area$ [ABC] = 10$ and $\tan (\angle ABC) = 5$. If the smallest possible value of $(\overline{AC})^2$ can be expressed as $-a + b\sqrt{c}$ for positive integers $a, b, c$, what is $a + b + c$?

2013 NIMO Problems, 3

Tags: geometry , angle bisector , pythagorean theorem

In triangle $ABC$, $AB=13$, $BC=14$ and $CA=15$. Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$. Find the smallest possible value of $n$. [i]Proposed by Evan Chen[/i]

2010 All-Russian Olympiad Regional Round, 11.3

Tags: geometry , perpendicular bisector , cyclic quadrilateral

Quadrangle $ABCD$ is inscribed in a circle with diameter $AC$. Points $K$ and $M$ are projections of vertices $A$ and $C$, respectively, onto line $BD$. A line parallel to $BC$ is drawn through point $K$ and intersecting $AC$ at point $P$. Prove that angle $KPM$ is a right angle.

1997 All-Russian Olympiad, 3

Tags: geometry

Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$. (You may assume that $C$ and $D$ lie on opposite sides of $A$.) [i]D. Tereshin[/i]

2024 Indonesia TST, 3

Tags: geometry

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Kvant 2019, M2564

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.

2020 Turkey Team Selection Test, 7

Tags: geometry

$A_1,A_2,B_1,B_2,C_1,C_2$ are points on a circle such that $A_1A_2 \parallel B_1B_2 \parallel C_1C_2 $ . $M$ is a point on same circle $MA_1$ and $B_2C_2$ intersect at $X$ , $MB_1$ and $A_2C_2$ intersect at $Y$, $MC_1$ and $A_2B_2$ intersect at $Z$ .Prove that $X , Y ,Z$ are collinear.

2018-2019 SDML (High School), 13

Tags: geometry , 3d geometry

A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube? [NEEDS DIAGRAM] $ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$

2022 Poland - Second Round, 2

Tags: circumcircle , perpendicular , geometry

Given a cyclic quadriteral $ABCD$. The circumcenter lies in the quadriteral $ABCD$. Diagonals $AC$ and $BD$ intersects at $S$. Points $P$ and $Q$ are the midpoints of $AD$ and $BC$. Let $p$ be a line perpendicular to $AC$ through $P$, $q$ perpendicular line to $BD$ through $Q$ and $s$ perpendicular to $CD$ through $S$. Prove that $p,q,s$ intersects at one point.

2000 Harvard-MIT Mathematics Tournament, 13

Tags: combinatorics , combinatorial geometry , geometry

Let $P_1, P_2,..., P_n$ be a convex $n$-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of $n$ obtained from strictly inside the polygon?

2019 ASDAN Math Tournament, 9

Tags: geometry

Consider triangle $\vartriangle ABC$ with circumradius $R = 10$, inradius $r = 2$ and semi-perimeter $S = 18$. Let $I$ be the incenter, and we extend $AI$, $BI$ and $CI$ to intersect the circumcircle at $D, E$ and $F$ respectively. Compute the area of $\vartriangle DEF$.

2019 Balkan MO Shortlist, G4

Tags: orthocenter , projective , geometry

Given an acute triangle $ABC$, let $M$ be the midpoint of $BC$ and $H$ the orthocentre. Let $\Gamma$ be the circle with diameter $HM$, and let $X,Y$ be distinct points on $\Gamma$ such that $AX,AY$ are tangent to $\Gamma$. Prove that $BXYC$ is cyclic.

2018 HMNT, 9

Tags: geometry

Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

1988 IMO Longlists, 88

Tags: geometry

Seven circles are given. That is, there are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two other adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, $A_1, A_2, A_3, A_4, A_5$ and $A_6$ prove that \[ A_1 A_2 \cdot A_3 A_4 \cdot A_5 A_6 = A_2 A_3 \cdot A_4 A_5 \cdot A_6 A_1. \]

2011 AMC 12/AHSME, 25

Tags: geometry , incenter , circumcircle , trigonometry , function , inequalities , calculus

Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? $\textbf{(A)}\ 60 ^\circ \qquad \textbf{(B)}\ 72 ^\circ\qquad \textbf{(C)}\ 75 ^\circ \qquad \textbf{(D)}\ 80 ^\circ\qquad \textbf{(E)}\ 90 ^\circ$

2023 MIG, 3

Tags: geometry , rectangle

A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle? $\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$

2016 Bundeswettbewerb Mathematik, 3

Tags: perpendicular , chord , geometry , circles

Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.

Ukrainian TYM Qualifying - geometry, 2011.5

Tags: geometry , locus , circumcenter , fixed

The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.

2021 Dutch IMO TST, 2

Tags: right angle , centroid , right triangle , geometry

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.