Olympiad problems

2009 IMO Shortlist, 8

Tags: geometry , circumcircle , trigonometry , inradius , incenter

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

2023 USA EGMO Team Selection Test, 3

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

2008 Harvard-MIT Mathematics Tournament, 26

Tags: parabola , conic , dilation , rotation , geometry , geometric transformation , ratio

Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.

2018 Swedish Mathematical Competition, 1

Tags: cyclic quadrilateral , geometry , parallel , angle bisector

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.

2023 Bulgarian Autumn Math Competition, 9.2

Tags: geometry

Given is an obtuse isosceles triangle $ABC$ with $CA=CB$ and circumcenter $O$. The point $P$ on $AB$ is such that $AP<\frac{AB} {2}$ and $Q$ on $AB$ is such that $BQ=AP$. The circle with diameter $CQ$ meets $(ABC)$ at $E$ and the lines $CE, AB$ meet at $F$. If $N$ is the midpoint of $CP$ and $ON, AB$ meet at $D$, show that $ODCF$ is cyclic.

2019 Sharygin Geometry Olympiad, 6

Tags: angle chasing , circumcircle , geometry , angle bisector , median

Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.

Novosibirsk Oral Geo Oly VIII, 2023.2

Tags: geometry , rectangle , square

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

1999 VJIMC, Problem 3

Tags: geometry

Let $A_1,\ldots,A_n$ be points of an ellipsoid with center $O$ in $\mathbb R^n$ such that $OA_i$, for $i=1,\ldots,n$, are mutually orthogonal. Prove that the distance of the point $O$ from the hyperplane $A_1A_2\ldots A_n$ does not depend on the choice of the points $A_1,\ldots,A_n$.

2020 Federal Competition For Advanced Students, P1, 2

Tags: right triangle , geometry , equal segments

Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$. (Walther Janous)

2019 Switzerland - Final Round, 7

Tags: angle , equal segments , equal angles , geometry

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

2020 Polish Junior MO First Round, 2.

Tags: easy , geometry

Points $P$ and $Q$ lie on the sides $AB$, $BC$ of the triangle $ABC$, such that $AC=CP =PQ=QB$ and $A \neq P$ and $C \neq Q$. If $\sphericalangle ACB = 104^{\circ}$, determine the measures of all angles of the triangle $ABC$.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , angle bisector

Let $ABC$ be an isosceles triangle such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$

2010 Junior Balkan Team Selection Tests - Romania, 4

Tags: isosceles , equal segments , angle , equal angles , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$. Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.

1957 Poland - Second Round, 3

Tags: 3d geometry , geometry

Given a cube with edge $ AB = a $ cm. Point $ M $ of segment $ AB $ is distant from the diagonal of the cube, which is oblique to $ AB $, by $ k $ cm. Find the distance of point $ M $ from the midpoint $ S $ of segment $ AB $.

2011 Indonesia TST, 3

Tags: geometry , equal segments

Let $\Gamma$ is a circle with diameter $AB$. Let $\ell$ be the tangent of $\Gamma$ at $A$, and $m$ be the tangent of $\Gamma$ through $B$. Let $C$ be a point on $\ell$, $C \ne A$, and let $q_1$ and $q_2$ be two lines that passes through $C$. If $q_i$ cuts $\Gamma$ at $D_i$ and $E_i$ ($D_i$ is located between $C$ and $E_i$) for $i = 1, 2$. The lines $AD_1, AD_2, AE_1, AE_2$ intersects $m$ at $M_1, M_2, N_1, N_2$ respectively. Prove that $M_1M_2 = N_1N_2$.

1995 Romania Team Selection Test, 3

Tags: geometry , sidelenghts , altitude , inradius

The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.

2012 Purple Comet Problems, 9

Tags: geometry , asymptote , rectangle

Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$. If $AB=11$ and $BC=8$, find the area of the quadrilateral $AEFB$.

2014 Sharygin Geometry Olympiad, 5

Tags: geometry , angle bisector

A triangle with angles of $30, 70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.) (A. Shapovalov )

2006 Princeton University Math Competition, 2

Tags: geometry

$ABC$ is an equilateral triangle with side length $ 1$. $BCDE$ is a square. Some point $F$ is equidistant from $A, D$, and $E$. Find the length of $AF$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/194318955f7ed5fed1c58633cb29c33011371a.jpg[/img]

Indonesia MO Shortlist - geometry, g2.3

Tags: geometry , ratio

For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that: \[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]

Swiss NMO - geometry, 2015.4

Tags: construction , circles , geometry

Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

2009 Sharygin Geometry Olympiad, 10

Tags: circumcircle , geometry , trapezoid , perpendicular bisector

Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$. Determine angle $ C$.

2020 Malaysia IMONST 1, 8

Tags: geometry , rectangle

Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17, PB = 15,$ and $PC = 6.$ What is the length of $PD$?

2023 USAJMO, 6

Tags: geometry

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

2004 Spain Mathematical Olympiad, Problem 5

Tags: median , inscribed circle , geometry

Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is: ${A/5 = B/10 = C/13}$.