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

2010 Contests, 3
Tags: geometry , power of a point , radical axis , geometry proposed
Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.

2008 Portugal MO, 2
Tags: geometry proposed , geometry
Let $AEBC$ be a cyclic quadrilateral. Let $D$ be a point on the ray $AE$ which is outside the circumscribed circumference of $AEBC$. Suppose that $\angle CAB=\angle BAE$. Prove that $AB=BD$ if and only if $DE=AC$.

2002 Tournament Of Towns, 3
Tags: geometry , geometry proposed
Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.

2010 Danube Mathematical Olympiad, 1
Tags: AMC , USA(J)MO , USAMO , geometry proposed , geometry
Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.

2001 Romania National Olympiad, 4
Tags: geometry , 3D geometry , geometry proposed
In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC),(A'CD),(A'DB)$ after the lines $d_1,d_2$ and $d_3$ respectively. a) Show that the lines $d_1,d_2,d_3$ intersect pairwise. b) Determine the area of the triangle formed by these three lines.

2005 Romania Team Selection Test, 3
Tags: geometry , 3D geometry , circumcircle , inequalities , tetrahedron , sphere , geometry proposed
Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$. [i]Calin Popescu[/i]

1996 Iran MO (3rd Round), 3
Tags: geometry proposed , geometry
Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.

2004 CentroAmerican, 2
Tags: geometry , trapezoid , circumcircle , angle bisector , geometry proposed
Let $ABCD$ be a trapezium such that $AB||CD$ and $AB+CD=AD$. Let $P$ be the point on $AD$ such that $AP=AB$ and $PD=CD$. $a)$ Prove that $\angle BPC=90^{\circ}$. $b)$ $Q$ is the midpoint of $BC$ and $R$ is the point of intersection between the line $AD$ and the circle passing through the points $B,A$ and $Q$. Show that the points $B,P,R$ and $C$ are concyclic.

2002 Iran MO (3rd Round), 19
Tags: geometry , incenter , geometry proposed
$I$ is incenter of triangle $ABC$. Incircle of $ABC$ touches $AB,AC$ at $X,Y$. $XI$ intersects incircle at $M$. Let $CM\cap AB=X'$. $L$ is a point on the segment $X'C$ that $X'L=CM$. Prove that $A,L,I$ are collinear iff $AB=AC$.

2004 Baltic Way, 19
Tags: geometry , circumcircle , geometric transformation , reflection , ratio , angle bisector , geometry proposed
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC$. Further, let $L$ be the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$, and let $K$ be the second intersection point of the circumcircle of the triangle $BAM$ with the side $AC$. Prove that $KL \parallel BC$.

2005 QEDMO 1st, 14 (G4)
Tags: trigonometry , geometry proposed , geometry
In the following, the abbreviation $g \cap h$ will mean the point of intersection of two lines $g$ and $h$. Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$. Prove that: \[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \] Darij

1987 Federal Competition For Advanced Students, P2, 1
Tags: geometry , angle bisector , geometry proposed
The sides $ a,b$ and the bisector of the included angle $ \gamma$ of a triangle are given. Determine necessary and sufficient conditions for such triangles to be constructible and show how to reconstruct the triangle.

2002 Baltic Way, 13
Tags: geometry , circumcircle , parallelogram , geometry proposed
Let $ABC$ be an acute triangle with $\angle BAC>\angle BCA$, and let $D$ be a point on side $AC$ such that $|AB|=|BD|$. Furthermore, let $F$ be a point on the circumcircle of triangle $ABC$ such that line $FD$ is perpendicular to side $BC$ and points $F,B$ lie on different sides of line $AC$. Prove that line $FB$ is perpendicular to side $AC$ .

2006 MOP Homework, 1
Tags: geometry , circumcircle , geometric transformation , reflection , trigonometry , perpendicular bisector , geometry proposed
$ ABC$ is an acute triangle. The points $ B'$ and $ C'$are the reflections of $ B$ and $ C$ across the lines $ AC$ and $ AB$ respectively. Suppose that the circumcircles of triangles$ ABB$' and $ ACC'$ meet at $ A$ and $ P$. Prove that the line $ PA$ passes through the circumcenter of triangle$ ABC.$

2011 Iran MO (3rd Round), 4
Tags: geometry , circumcircle , incenter , geometric transformation , reflection , radical axis , geometry proposed
A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter. [i]proposed by Masoud Nourbakhsh[/i]

1984 Iran MO (2nd round), 7
Tags: geometry proposed , geometry
Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)

2007 Romania Team Selection Test, 2
Tags: geometry , circumcircle , geometry proposed
Let $ABC$ be a triangle, $E$ and $F$ the points where the incircle and $A$-excircle touch $AB$, and $D$ the point on $BC$ such that the triangles $ABD$ and $ACD$ have equal in-radii. The lines $DB$ and $DE$ intersect the circumcircle of triangle $ADF$ again in the points $X$ and $Y$. Prove that $XY\parallel AB$ if and only if $AB=AC$.

1986 Balkan MO, 1
Tags: geometry , incenter , circumcircle , geometry proposed
A line passing through the incenter $I$ of the triangle $ABC$ intersect its incircle at $D$ and $E$ and its circumcircle at $F$ and $G$, in such a way that the point $D$ lies between $I$ and $F$. Prove that: $DF \cdot EG \geq r^{2}$.

2002 Turkey Team Selection Test, 2
Tags: ratio , geometry , angle bisector , geometry proposed
In a triangle $ABC$, the angle bisector of $\widehat{ABC}$ meets $[AC]$ at $D$, and the angle bisector of $\widehat{BCA}$ meets $[AB]$ at $E$. Let $X$ be the intersection of the lines $BD$ and $CE$ where $|BX|=\sqrt 3|XD|$ ve $|XE|=(\sqrt 3 - 1)|XC|$. Find the angles of triangle $ABC$.

2014 District Olympiad, 3
Tags: geometry proposed , geometry
The medians $AD, BE$ and $CF$ of triangle $ABC$ intersect at $G$. Let $P$ be a point lying in the interior of the triangle, not belonging to any of its medians. The line through $P$ parallel to $AD$ intersects the side $BC$ at $A_{1}$. Similarly one defines the points $B_{1}$ and $C_{1}$. Prove that \[ \overline{A_{1}D}+\overline{B_{1}E}+\overline{C_{1}F}=\frac{3}{2}\overline{PG} \]

2009 Baltic Way, 11
Tags: geometry , geometric transformation , homothety , angle bisector , geometry proposed
Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$, and let $K$ be a point on the ray $BA$ beyond $A$. The line $KM$ intersects the side $BC$ at the point $L$. $P$is the point on the segment $BM$ such that $PM$ is the bisector of the angle $LPK$. The line $\ell$ passes through $A$ and is parallel to $BM$. Prove that the projection of the point $M$ onto the line $\ell$ belongs to the line $PK$.

2009 District Round (Round II), 4
Tags: geometry , circumcircle , parallelogram , geometry proposed
in an acute triangle $ABC$,$D$ is a point on $BC$,let $Q$ be the intersection of $AD$ and the median of $ABC$from $C$,$P$ is a point on $AD$,distinct from $Q$.the circumcircle of $CPD$ intersects $CQ$ at $C$ and $K$.prove that the circumcircle of $AKP$ passes through a fixed point differ from $A$.

2008 Polish MO Finals, 3
Tags: trigonometry , geometry proposed , geometry
In a convex pentagon $ ABCDE$ in which $ BC\equal{}DE$ following equalities hold: \[ \angle ABE \equal{}\angle CAB \equal{}\angle AED\minus{}90^{\circ},\qquad \angle ACB\equal{}\angle ADE\] Show that $ BCDE$ is a parallelogram.

2010 Polish MO Finals, 1
Tags: inequalities , geometry , perimeter , parallelogram , geometry proposed
On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that \[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]

2006 Moldova Team Selection Test, 2
Tags: geometry , circumcircle , power of a point , radical axis , geometry proposed
Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.