Olympiad problems

2009 Iran MO (3rd Round), 5

Tags: geometry

5-Two circles $ S_1$ and $ S_2$ with equal radius and intersecting at two points are given in the plane.A line $ l$ intersects $ S_1$ at $ B,D$ and $ S_2$ at $ A,C$(the order of the points on the line are as follows:$ A,B,C,D$).Two circles $ W_1$ and $ W_2$ are drawn such that both of them are tangent externally at $ S_1$ and internally at $ S_2$ and also tangent to $ l$ at both sides.Suppose $ W_1$ and $ W_2$ are tangent.Then PROVE $ AB \equal{} CD$.

2004 Vietnam National Olympiad, 2

Tags: angle bisector , geometry

In a triangle $ ABC$, the bisector of $ \angle ACB$ cuts the side $ AB$ at $ D$. An arbitrary circle $ (O)$ passing through $ C$ and $ D$ meets the lines $ BC$ and $ AC$ at $ M$ and $ N$ (different from $ C$), respectively. (a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$. (b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.

2007 ITest, 60

Tags: circumcircle , geometry

Let $T=\text{TNFTPP}$. Triangle $ABC$ has $AB=6T-3$ and $AC=7T+1$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A$, $B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A$, $C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B'$, $C'$, $O_1$, and $O_2$ lie on a circle, find the length of $BC$.

1990 China National Olympiad, 1

Tags: geometry

Given a convex quadrilateral $ABCD$, side $AB$ is not parallel to side $CD$. The circle $O_1$ passing through $A$ and $B$ is tangent to side $CD$ at $P$. The circle $O_2$ passing through $C$ and $D$ is tangent to side $AB$ at $Q$. Circle $O_1$ and circle $O_2$ meet at $E$ and $F$. Prove that $EF$ bisects segment $PQ$ if and only if $BC\parallel AD$.

2015 India IMO Training Camp, 1

Tags: geometric transformation , circumcircle , geometry , reflection

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

1989 Tournament Of Towns, (214) 2

Tags: tangential , geometry , trapezoid , incircle , tangent circle

It is known that a circle can be inscribed in a trapezium $ABCD$. Prove that the two circles, constructed on its oblique sides as diameters, touch each other. (D. Fomin, Leningrad)

2014 China Team Selection Test, 1

Tags: geometry , perpendicular bisector , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.

2003 National High School Mathematics League, 1

Tags: geometry

Draw two tangents to the circle from point $P$ outside a circle, touching the circle at $A$ and $B$, then draw a secant line passes $P$, intersecting the circle at points $C$ and $D$ ($C$ is between $P$ and $D$). $Q$ is a point on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

2024 Sharygin Geometry Olympiad, 9.4

Tags: geometry , combi geo , geo

For which $n > 0$ it is possible to mark several different points and several different circles on the plane in such a way that: — exactly $n$ marked circles pass through each marked point; — exactly $n$ marked points lie on each marked circle; — the center of each marked circle is marked?

1994 Poland - First Round, 8

Tags: geometry , pyramid , 3d geometry

In a regular pyramid with a regular $n$-gon as a base, the dihedral angle between a lateral face and the base is equal to $\alpha$, and the angle between a lateral edge and the base is equal to $\beta$. Prove that $sin^2 \alpha - sin^2 \beta \leq tg^2 \frac{\pi}{2n}$.

1975 Bundeswettbewerb Mathematik, 2

Tags: combinatorial geometry , geometry , polyhedron , 3d geometry , edge

Prove that in each polyhedron there exist two faces with the same number of edges.

2020 Stanford Mathematics Tournament, 6

Tags: analytic geometry , geometry

Consider triangle $ABC$ on the coordinate plane with $A = (2, 3)$ and $C =\left( \frac{96}{13} , \frac{207}{13} \right)$. Let $B$ be the point with the smallest possible $y$-coordinate such that $AB = 13$ and $BC = 15$. Compute the coordinates of the incenter of triangle $ABC$.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.1

Tags: algebra , geometry

Let's call this position of the hour and minute hands on the analog clock [i]wonderful[/i], during which the hands change places after some time. Count the total number of wonderful clockwise positions.

2009 Germany Team Selection Test, 2

Tags: homothety , trigonometry , circumcircle , geometry , quadrilateral , inversion

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

1998 Bundeswettbewerb Mathematik, 3

Tags: geometry , complex number

Let F be the midpoint of side BC or triangle ABC. Construct isosceles right triangles ABD and ACE externally on sides AB and AC with the right angles at D and E respectively. Show that DEF is an isosceles right triangle.

2016 Mathematical Talent Reward Programme, SAQ: P 4

Tags: geometry

For any given $k$ points in a plane, we define the diameter of the points as the maximum distance between any two points among the given points. Suppose $n$ points are there in a plane with diameter $d$. Show that we can always find a circle with radius $\frac{\sqrt{3}}{2}d$ such that all points lie inside the circle.

1996 Yugoslav Team Selection Test, Problem 2

Tags: combinatorics , geometry , combinatorial geometry

Let there be given a set of $1996$ equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.

2009 Belarus Team Selection Test, 3

Tags: tangential , trapezoid , geometry , right angle , parallel , tangential quadrilateral

Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$. I. Voronovich

2000 AIME Problems, 12

Tags: number theory , geometry , trigonometry , circumcircle , 3d geometry , sphere , tetrahedron

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

1994 Tournament Of Towns, (437) 3

Tags: geometry , angle , median

The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

Kharkiv City MO Seniors - geometry, 2016.11.5

Tags: equal segments , circles , geometry , circumcenter

The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.

2017 All-Russian Olympiad, 8

Tags: geometry , incenter , angle chasing , incircle

Given a convex quadrilateral $ABCD$. We denote $I_A,I_B, I_C$ and $I_D$ centers of $\omega_A, \omega_B,\omega_C $and $\omega_D$,inscribed In the triangles $DAB, ABC, BCD$ and $CDA$, respectively.It turned out that $\angle BI_AA + \angle I_CI_AI_D = 180^\circ$. Prove that $\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}$. (A. Kuznetsov)

Indonesia MO Shortlist - geometry, g6.7

Tags: ratio , inradius , geometry

Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to $ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$ (hmm,, looks familiar, isn't it? :wink: )

2006 Silk Road, 4

Tags: induction , geometry

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$. Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: [b]1)[/b] The line $l$ is bounding the line $l'$; [b]2)[/b] the line $l'$ is not bounding the line $l$.

2010 Contests, 2

Tags: cyclic quadrilateral , circumcircle , geometry

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .