This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2023

2013 APMO, 5
Tags: ratio , geometry , circumcircle , geometry proposed
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.

2004 ITAMO, 2
Tags: geometry , geometric transformation , homothety , geometry proposed
Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

2001 Romania Team Selection Test, 2
Tags: symmetry , geometry proposed , geometry
The vertices $A,B,C$ and $D$ of a square lie outside a circle centred at $M$. Let $AA',BB',CC',DD'$ be tangents to the circle. Assume that the segments $AA',BB',CC',DD'$ are the consecutive sides of a quadrilateral $p$ in which a circle is inscribed. Prove that $p$ has an axis of symmetry.

1990 IberoAmerican, 4
Tags: conics , parabola , geometry , rectangle , power of a point , radical axis , geometry proposed
Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$. a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies. b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.

2014 Uzbekistan National Olympiad, 4
Tags: trigonometry , geometry proposed , geometry
A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $. Find the angle $ \angle BDC$.

2014 India Regional Mathematical Olympiad, 1
Tags: geometry , trapezoid , similar triangles , geometry proposed
In acute $\triangle ABC,$ let $D$ be the foot of perpendicular from $A$ on $BC$. Consider points $K, L, M$ on segment $AD$ such that $AK= KL= LM= MD$. Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that $BD= DC$. [img]http://s27.postimg.org/a0d0plr4z/Untitled.png[/img]

1993 Romania Team Selection Test, 2
Tags: geometry proposed , geometry
Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,R)$ and circumscribed to the circle $\mathcal{C}(L,r)$. Denote $d=\dfrac{Rr}{R+r}$. Show that there exists a triangle $DEF$ such that for any interior point $M$ in $ABC$ there exists a point $X$ on the sides of $DEF$ such that $MX\le d$. [i]Dan Brânzei[/i]

2010 Turkey Team Selection Test, 1
Tags: geometry , incenter , circumcircle , trigonometry , trig identities , Law of Sines , geometry proposed
$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$

2010 Turkey MO (2nd round), 1
Tags: geometry , circumcircle , geometry proposed
Let $A$ and $B$ be two points on the circle with diameter $[CD]$ and on the different sides of the line $CD.$ A circle $\Gamma$ passing through $C$ and $D$ intersects $[AC]$ different from the endpoints at $E$ and intersects $BC$ at $F.$ The line tangent to $\Gamma$ at $E$ intersects $BC$ at $P$ and $Q$ is a point on the circumcircle of the triangle $CEP$ different from $E$ and satisfying $|QP|=|EP|. \: AB \cap EF =\{R\}$ and $S$ is the midpoint of $[EQ].$ Prove that $DR$ is parallel to $PS.$

2012 Spain Mathematical Olympiad, 3
Tags: circumcircle , geometry proposed , geometry
Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.

2011 Stars Of Mathematics, 1
Tags: geometry , circumcircle , geometry proposed
Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram. Prove the lines $AC$, $KH$, $BD$ are concurrent. (I. Nagel)

2006 JBMO ShortLists, 9
Tags: geometry , trapezoid , geometry proposed
Let $ ABCD$ be a trapezoid with $ AB\parallel CD,AB>CD$ and $ \angle{A} \plus{} \angle{B} \equal{} 90^\circ$. Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.

2010 IberoAmerican, 3
Tags: geometry , circumcircle , incenter , projective geometry , geometry proposed
The circle $ \Gamma $ is inscribed to the scalene triangle $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $ in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $) be the intersections of $ \Gamma $ with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent. [i]Author: Arnoldo Aguilar, El Salvador[/i]

2002 District Olympiad, 2
Tags: trigonometry , geometry proposed , geometry
In the $xOy$ system, consider the points $A_n(n,n^3)$ with $n\in \mathbb{N}^*$ and the point $B(0,1)$. Prove that a) for any positive integers $k>j>i\ge 1$, the points $A_i,A_j,A_k$ cannot be collinear. b) for any positive integers $i_k>i_{k-1}>\ldots>i_1\ge 1$, we have \[\mu(\widehat{A_{i_1}OB})+\mu(\widehat{A_{i_2}OB})+\cdots+\mu(\widehat{A_{i_k}OB})<\frac{\pi}{2}\] [i]***[/i]

1998 Baltic Way, 12
Tags: geometry , circumcircle , geometry proposed
In a triangle $ABC$, $\angle BAC =90^{\circ}$. Point $D$ lies on the side $BC$ and satisfies $\angle BDA=2\angle BAD$. Prove that \[\frac{2}{AD}=\frac{1}{BD}+\frac{1}{CD} \]

1999 Brazil National Olympiad, 6
Tags: geometry , circumcircle , geometric transformation , rotation , geometry proposed , Brocard , Miquel point
Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.

2004 Iran MO (3rd Round), 11
Tags: geometry , circumcircle , projective geometry , modular arithmetic , Euler , geometry proposed
assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle at A" in the point $X_a$; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.

2011 Baltic Way, 13
Tags: geometry proposed , geometry , similarity , Isogonal conjugate
Let $E$ be an interior point of the convex quadrilateral $ABCD$. Construct triangles $\triangle ABF,\triangle BCG,\triangle CDH$ and $\triangle DAI$ on the outside of the quadrilateral such that the similarities $\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE$ and $ \triangle DAI\sim\triangle CBE$ hold. Let $P,Q,R$ and $S$ be the projections of $E$ on the lines $AB,BC,CD$ and $DA$, respectively. Prove that if the quadrilateral $PQRS$ is cyclic, then \[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]

1993 Balkan MO, 3
Tags: geometry , incenter , circumcircle , geometry proposed
Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$, respectively, are externally tangent at point $\lambda$. A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$, $O_2$ lie inside $C$. The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$. If $D$ is the midpoint of the segment $KL$, show that $\angle O_1OO_2 = \angle ADB$. [i]Greece[/i]

2008 China Team Selection Test, 1
Tags: geometry , circumcircle , perpendicular bisector , power of a point , radical axis , geometry proposed
Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

1986 IMO Longlists, 80
Tags: geometry , 3D geometry , tetrahedron , incenter , geometry proposed
Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$

2010 Contests, 1
Tags: complex numbers , geometry proposed , geometry
Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

2013 Mediterranean Mathematics Olympiad, 4
Tags: geometry proposed , geometry
$ABCD$ is quadrilateral inscribed in a circle $\Gamma$ .Lines $AB$ and $CD$ intersect at $E$ and lines$AD$ and $BC$ intersect at $F$. Prove that the circle with diameter $EF$ and circle $\Gamma$ are orthogonal.

2003 Iran MO (3rd Round), 18
Tags: geometry , 3D geometry , tetrahedron , circumcircle , sphere , geometry proposed
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.

2004 Iran MO (3rd Round), 27
Tags: Support , vector , geometry proposed , geometry
$ \Delta_1,\ldots,\Delta_n$ are $ n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.