Olympiad problems

2014 South africa National Olympiad, 3

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

2005 Postal Coaching, 16

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

The diagonals AC and BD of a cyclic ABCD intersect at E. Let O be circumcentre of ABCD. If midpoints of AB, CD, OE are collinear prove that AD=BC. Bomb [color=red][Moderator edit: The problem is wrong. See also http://www.mathlinks.ro/Forum/viewtopic.php?t=53090 .][/color]

2023 Sharygin Geometry Olympiad, 10.5

Tags: geometry

The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $M$ be the midpoint of arc $\widehat{BAC}$ of the circumcircle, and $P$, $Q$ be the projections of $M$ to the external bisectors of angles $B$ and $C$ respectively. Prove that the line $PQ$ bisects $AD$.

1999 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , geometry , rectangle , trigonometry

A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

2015 India Regional MathematicaI Olympiad, 8

Tags: inequalities , number theory , geometry

The length of each side of a convex quadrilateral $ABCD$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.

2018 Bosnia and Herzegovina Junior BMO TST, 3

Tags: geometry

Let $\Gamma$ be circumscribed circle of triangle $ABC $ $(AB \neq AC)$. Let $O$ be circumcenter of the triangle $ABC$. Let $M$ be a point where angle bisector of angle $BAC$ intersects $\Gamma$. Let $D$ $(D \neq M)$ be a point where circumscribed circle of the triangle $BOM$ intersects line segment $AM$ and let $E$ $(E \neq M)$ be a point where circumscribed circle of triangle $COM$ intersects line segment $AM$. Prove that $BD+CE=AM$.

1998 Iran MO (3rd Round), 2

Tags: reflection , trigonometry , circumcircle , isogonal conjugate , geometry

Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that \[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC. \] Prove that \[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1. \]

2007 AMC 12/AHSME, 2

Tags: geometry

An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. It is filled with water to a height of $ 40$ cm. A brick with a rectangular base that measures $ 40$ cm by $ 20$ cm and a height of $ 10$ cm is placed in the aquarium. By how many centimeters does the water rise? $ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$

2009 Canadian Mathematical Olympiad Qualification Repechage, 2

Tags: pythagorean theorem , trigonometry , geometry

Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$

Cono Sur Shortlist - geometry, 2012.G3

Tags: circumcircle , power of a point , radical axis , geometry

Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.

2003 Romania Team Selection Test, 11

Tags: trigonometry , pigeonhole principle , area of a triangle , geometry

In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21. [i]Laurentiu Panaitopol[/i]

2024 India IMOTC, 24

Tags: combinatorics , geometry , combinatorial geometry

There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that [color=#FFFFFF]___[/color]$\bullet$ Each circle in $\mathcal C$ has unit radius. [color=#FFFFFF]___[/color]$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$. [color=#FFFFFF]___[/color]$\bullet$ There are less than $0.3n$ pairs of circles in $\mathcal C$ that intersect in exactly $2$ points. [i]Note: Weaker results with $\it{0.3n}$ replaced by $\it{cn}$ may be awarded points depending on the value of the constant $\it{c > 0.3}$.[/i] [i]Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param[/i]

2006 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , combinatorial geometry , combinatorics , triangle area , area , trapezoid

In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.

2013 Romania Team Selection Test, 2

Tags: geometry

Let $K$ be a convex quadrangle and let $l$ be a line through the point of intersection of the diagonals of $K$. Show that the length of the segment of intersection $l\cap K$ does not exceed the length of (at least) one of the diagonals of $K$.

1957 Poland - Second Round, 6

Tags: tangential , geometry

Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e. an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other.

2009 District Olympiad, 2

Tags: geometry , geometric inequality

Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$

1988 Mexico National Olympiad, 6

Tags: geometry , locus , incenter , circles

Consider two fixed points $B,C$ on a circle $w$. Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$.

1989 IMO Longlists, 65

Tags: geometry

Let $ ABCD$ be a quadrilateral inscribed in a circle of radius $ AB$ such that $ BC \equal{} a, CD \equal{} b,$ $ DA \equal{} \frac{3 \sqrt{3} \minus{} 1}{2} \cdot a$ For each point $ M$ on the semicircle with radius $ AB$ not containing $ C$ and $ D,$ denote by $ h_1, h_2, h_3$ the distances from $ M$ to the straight lines (sides) $ BC, CD,$ and $ DA.$ Find the maximum of $ h_1 \plus{} h_2 \plus{} h_3.$

2024 Korea - Final Round, P4

Tags: geometry , circumcircle , pascal

For a triangle $ABC$, $O$ is the circumcircle and $D$ is a point on ray $BA$. $E$ and $F$ are points on $O$ so that $DE$ and $DF$ are tangent to $O$ and $EF$ cuts $AC$ at $T(\neq C)$. $P(\neq B,C)$ is a point on the arc $BC$ not containing $A$, and $DP$ cuts $O$ at $Q (\neq P)$. Let $BQ$ and $DT$ meets on $X (\neq Q)$, and $PT$ cuts $O$ at $Y (\neq P)$. Prove that $C,X,Y$ are collinear.

2000 Stanford Mathematics Tournament, 22

Tags: geometry

An equilateral triangle with sides of length $4$ has an isosceles triangle with the same base and half the height cut out of it. Find the remaining area

2017 Hanoi Open Mathematics Competitions, 11

Tags: square , geometry

Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

2021 Polish Junior MO Second Round, 2

Tags: geometry , right angle , square

Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.

1994 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: geometry , 3d geometry , rotation , symmetry

In how many ways can you color the six sides of a cube in black or white? (Do note that the cube is unchanged when rotated?) A. 7 B. 10 C. 20 D. 30 E. 36

2004 Germany Team Selection Test, 2

Tags: locus problems , geometric inequalities , geometry

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

1997 APMO, 3

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

Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.