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

2003 India IMO Training Camp, 8
Tags: geometry , inradius , geometry proposed
Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.

2014 Moldova Team Selection Test, 3
Tags: geometry , circumcircle , trapezoid , geometry proposed
Let $ABCD$ be a cyclic quadrilateral. The bisectors of angles $BAD$ and $BCD$ intersect in point $K$ such that $K \in BD$. Let $M$ be the midpoint of $BD$. A line passing through point $C$ and parallel to $AD$ intersects $AM$ in point $P$. Prove that triangle $\triangle DPC$ is isosceles.

2012 Bulgaria National Olympiad, 3
Tags: geometry , circumcircle , geometry proposed
We are given an acute-angled triangle $ABC$ and a random point $X$ in its interior, different from the centre of the circumcircle $k$ of the triangle. The lines $AX,BX$ and $CX$ intersect $k$ for a second time in the points $A_1,B_1$ and $C_1$ respectively. Let $A_2,B_2$ and $C_2$ be the points that are symmetric of $A_1,B_1$ and $C_1$ in respect to $BC,AC$ and $AB$ respectively. Prove that the circumcircle of the triangle $A_2,B_2$ and $C_2$ passes through a constant point that does not depend on the choice of $X$.

2016 Croatia Team Selection Test, Problem 3
Tags: geometry , circumcircle , incircle , geometry proposed
Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

2007 Iran Team Selection Test, 1
Tags: geometry , trigonometry , circumcircle , angle bisector , geometry proposed
In triangle $ABC$, $M$ is midpoint of $AC$, and $D$ is a point on $BC$ such that $DB=DM$. We know that $2BC^{2}-AC^{2}=AB.AC$. Prove that \[BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}\]

1992 Balkan MO, 3
Tags: inequalities , geometry , circumcircle , trigonometry , geometry proposed
Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \] [i]Greece[/i]

Swiss NMO - geometry, 2010.9
Tags: geometry , circumcircle , power of a point , radical axis , geometry proposed
Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$. Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.

2012 ELMO Shortlist, 2
Tags: geometry , circumcircle , power of a point , radical axis , cyclic quadrilateral , geometry proposed
In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$. a) Prove $SX,TY, AD$ are concurrent at a point $Z$. b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$. [i]Ray Li.[/i]

1990 IMO Longlists, 34
Tags: geometry proposed , geometry
There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.

2009 Korea National Olympiad, 2
Tags: geometry , incenter , geometry proposed
Let $ABC$ be a triangle and $ P, Q ( \ne A, B, C ) $ are the points lying on segments $ BC , CA $. Let $ I, J, K $ be the incenters of triangle $ ABP, APQ, CPQ $. Prove that $ PIJK $ is a convex quadrilateral.

2009 China Team Selection Test, 1
Tags: geometry , geometry proposed
In convex pentagon $ ABCDE$, denote by $ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command. = B',CF%Error. "capDB" is a bad command. = C',DG\cap EC = D',EH\cap AD = E'.$ Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.

1993 All-Russian Olympiad, 2
Tags: geometry , parallelogram , geometry proposed
Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$.

2010 Indonesia TST, 3
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 $T$.

2013 Iran MO (2nd Round), 1
Tags: geometry , circumcircle , geometric transformation , reflection , geometry proposed
Let $P$ be a point out of circle $C$. Let $PA$ and $PB$ be the tangents to the circle drawn from $C$. Choose a point $K$ on $AB$ . Suppose that the circumcircle of triangle $PBK$ intersects $C$ again at $T$. Let ${P}'$ be the reflection of $P$ with respect to $A$. Prove that \[ \angle PBT = \angle {P}'KA \]

2011 Romania Team Selection Test, 2
Tags: geometry , power of a point , radical axis , geometry proposed
Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

2015 International Zhautykov Olympiad, 2
Tags: geometry , symmetry , trigonometry , geometry proposed
Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $ meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.

2002 JBMO ShortLists, 13
Tags: inequalities , geometry , rectangle , geometry proposed
Let $ A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $ 1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $ M,N,P$ of the rectangle such that $ MA_1 + MA_2 + \cdots + MA_{2002} + $ $NA_1 + NA_2 + \cdots + NA_{2002} + $ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$.

2010 Contests, 3
Tags: geometry , geometry proposed
On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

2010 Romania National Olympiad, 4
Tags: search , geometry , similar triangles , geometry proposed
On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$. [i]Nicolae Bourbacut[/i]

1999 Greece National Olympiad, 3
Tags: geometry , circumcircle , geometry proposed
In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.

2005 JBMO Shortlist, 1
Tags: geometry , power of a point , radical axis , geometry proposed
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

2007 China Western Mathematical Olympiad, 2
Tags: geometry , circumcircle , power of a point , radical axis , geometry proposed
Let $ C$ and $ D$ be two intersection points of circle $ O_1$ and circle $ O_2$. A line, passing through $ D$, intersects the circle $ O_1$ and the circle $ O_2$ at the points $ A$ and $ B$ respectively. The points $ P$ and $ Q$ are on circles $ O_1$ and $ O_2$ respectively. The lines $ PD$ and $ AC$ intersect at $ H$, and the lines $ QD$ and $ BC$ intersect at $ M$. Suppose that $ O$ is the circumcenter of the triangle $ ABC$. Prove that $ OD\perp MH$ if and only if $ P,Q,M$ and $ H$ are concyclic.

2015 Macedonia National Olympiad, Problem 1
Tags: geometry proposed , geometry
Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.

2014 Baltic Way, 15
Tags: inequalities , triangle inequality , geometry proposed , geometry
The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that \[AB \cdot CD + AD \cdot BC < AC(AB + AD).\]

2000 Turkey MO (2nd round), 2
Tags: geometry proposed , geometry
A positive real number $a$ and two rays wich intersect at point $A$ are given. Show that all the circles which pass through $A$ and intersect these rays at points $B$ and $C$ where $|AB|+|AC|=a$ have a common point other than $A$.