Olympiad problems

2021 Austrian Junior Regional Competition, 2

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

1995 Abels Math Contest (Norwegian MO), 2a

Tags: perpendicular , tangent circle , circles , geometry

Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively. Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.

1968 Poland - Second Round, 2

Tags: construction , geometry , geometric construction , orthocenter

Given a circle $ k $ and a point inside it $ H $. Inscribe a triangle in the circle such that this point $ H $ is the point of intersection of the triangle's altitudes.

2006 MOP Homework, 5

Tags: geometry

Show that among the vertices of any area $1$ convex polygon with $n > 3$ sides there exist four such that the quadrilateral formed by these four has area at least $1/2$.

2003 France Team Selection Test, 3

Tags: circumcircle , trig identities , incenter , geometry , trigonometry , law of sines

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

2015 Sharygin Geometry Olympiad, P18

Tags: collinear , hexagon , geometry

Let $ABCDEF$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $AB$ and $CD$, $AC$ and $BD$, $AF$ and $DE$, $AE$ and $DF$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.

1976 IMO Longlists, 44

Tags: floor function , geometry

A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?

1969 All Soviet Union Mathematical Olympiad, 127

Tags: geometry , geometric inequality

Let $h_k$ be an apothem of the regular $k$-gon inscribed into a circle with radius $R$. Prove that $$(n + 1)h_{n+1} - nh_n > R$$

2016 India Regional Mathematical Olympiad, 3

Tags: geometry

Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.

2015 Turkey Team Selection Test, 4

Tags: geometry , inversion , tangent

Let $ABC$ be a triangle such that $|AB|=|AC|$ and let $D,E$ be points on the minor arcs $\overarc{AB}$ and $\overarc{AC}$ respectively. The lines $AD$ and $BC$ intersect at $F$ and the line $AE$ intersects the circumcircle of $\triangle FDE$ a second time at $G$. Prove that the line $AC$ is tangent to the circumcircle of $\triangle ECG$.

2008 Mexico National Olympiad, 2

Tags: number theory , 3d geometry , greatest common divisor , geometry

We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices. a) Prove that $\frac23E\le V$. b) Can $E=V$?

2010 India National Olympiad, 5

Tags: geometric transformation , circumcircle , geometry , euler , perpendicular bisector , reflection

Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.

2011 AMC 8, 2

Tags: geometry

Karl's rectangular vegetable garden is $20$ by $45$ feet, and Makenna's is $25$ by $40$ feet. Which garden is larger in area? $\textbf{(A)}$ Karl's garden is larger by 100 square feet. $\textbf{(B)}$ Karl's garden is larger by 25 square feet. $\textbf{(C)}$ The gardens are the same size. $\textbf{(D)}$ Makenna's garden is larger by 25 square feet. $\textbf{(E)}$ Makenna's garden is larger by 100 square feet.

2022 Taiwan Mathematics Olympiad, 5

Tags: geometry

Let $J$ be the $A$-excenter of an acute triangle $ABC$. Let $X$, $Y$ be two points on the circumcircle of the triangle $ACJ$ such that $\overline{JX} = \overline{JY} < \overline{JC}$. Let $P$ be a point lies on $XY$ such that $PB$ is tangent to the circumcircle of the triangle $ABC$. Let $Q$ be a point lies on the circumcircle of the triangle $BXY$ such that $BQ$ is parallel to $AC$. Prove that $\angle BAP = \angle QAC$. [i]Proposed by Li4.[/i]

2021 Sharygin Geometry Olympiad, 10-11.8

Tags: geometry , locus , arc

On the attraction "Merry parking", the auto has only two position* of a steering wheel: "right", and "strongly right". So the auto can move along an arc with radius $r_1$ or $r_2$. The auto started from a point $A$ to the Nord, it covered the distance $\ell$ and rotated to the angle $a < 2\pi$. Find the locus of its possible endpoints.

2021 JHMT HS, 6

Tags: angle bisector , geometry

Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$

2008 Costa Rica - Final Round, 2

Tags: geometry , angle bisector , circumcircle , projective geometry

Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.

2023 All-Russian Olympiad, 4

Tags: geometry

Let $\omega$ be the circumcircle of triangle $ABC$ with $AB<AC$. Let $I$ be its incenter and let $M$ be the midpoint of $BC$. The foot of the perpendicular from $M$ to $AI$ is $H$. The lines $MH, BI, AB$ form a triangle $T_b$ and the lines $MH, CI, AC$ form a triangle $T_c$. The circumcircle of $T_b$ meets $\omega$ at $B'$ and the circumcircle of $T_c$ meets $\omega$ at $C'$. Prove that $B', H, C'$ are collinear.

2018 District Olympiad, 3

Tags: 3d geometry , cube , parallelepiped , collinear , geometry

Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.

1974 IMO Longlists, 15

Tags: circumcircle , geometry , trigonometry , triangle , trigonometric inequality

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

2016 Peru IMO TST, 12

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2023 Princeton University Math Competition, A4 / B6

Tags: geometry

Let $\vartriangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\vartriangle ABC$ again at the point $M\ne A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at two points, $X$ and $Y$ . Compute $AX \cdot AY$.

2005 USAMTS Problems, 5

Tags: geometry , 3d geometry , sphere

Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.

2016 Iran Team Selection Test, 4

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2018-2019 Fall SDPC, 7

Tags: geometry

The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.