Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
complex numbers
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
prove triangles are similar
N.T.TUAN   58
N Monday at 2:55 PM by mathwiz_1207
Source: USA Team Selection Test 2007, Problem 5
Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T = BT = C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.
58 replies
N.T.TUAN
Dec 8, 2007
mathwiz_1207
Monday at 2:55 PM
J
H
RMM 2013 Problem 6
dr_Civot   15
N Monday at 8:09 AM by N3bula
A token is placed at each vertex of a regular $2n$-gon. A move consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
15 replies
dr_Civot
Mar 3, 2013
N3bula
Monday at 8:09 AM
J
H
incircle excenter midpoints
danepale   9
N May 18, 2025 by Want-to-study-in-NTU-MATH
Source: Middle European Mathematical Olympiad T-6
Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.

Prove that the points $B, C, N,$ and $L$ are concyclic.
9 replies
danepale
Sep 21, 2014
Want-to-study-in-NTU-MATH
May 18, 2025
J
H
Three mutually tangent circles
math154   8
N May 17, 2025 by lakshya2009
Source: ELMO Shortlist 2011, G2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.

David Yang.
8 replies
math154
Jul 3, 2012
lakshya2009
May 17, 2025
J
H
Line AT passes through either S_1 or S_2
v_Enhance   89
N May 17, 2025 by zuat.e
Source: USA December TST for 57th IMO 2016, Problem 2
Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$.

Proposed by Evan Chen
89 replies
v_Enhance
Dec 21, 2015
zuat.e
May 17, 2025
J
H
Eight-point cicle
sandu2508   15
N May 16, 2025 by Mamadi
Source: Balkan MO 2010, Problem 2
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$.
Prove that $M_1$ lies on the segment $BH_1$.
15 replies
sandu2508
May 4, 2010
Mamadi
May 16, 2025
J
H
RMM 2013 Problem 3
dr_Civot   79
N May 16, 2025 by Ilikeminecraft
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
79 replies
dr_Civot
Mar 2, 2013
Ilikeminecraft
May 16, 2025
J
H
Hard Inequality
Asilbek777   2
N May 14, 2025 by Ritwin
Waits for Solution
2 replies
Asilbek777
May 14, 2025
Ritwin
May 14, 2025
J
H
ISI UGB 2025 P7
SomeonecoolLovesMaths   12
N May 14, 2025 by ohiorizzler1434
Source: ISI UGB 2025 P7
Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

IMAGE
12 replies
SomeonecoolLovesMaths
May 11, 2025
ohiorizzler1434
May 14, 2025
J
H
angles in triangle
AndrewTom   34
N May 13, 2025 by happypi31415
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
34 replies
AndrewTom
Feb 1, 2013
happypi31415
May 13, 2025
J
a