G
Topic
First Poster
Last Poster
Tangents inducing isogonals
nikolapavlovic 55
N
by SimplisticFormulas
Source: Serbian MO 2017 6
Let 
be the circumcircle of 
and let 
be A-excircle .Let the two common tangents of 
cut 
in 
.Prove that 
.
be the circumcircle of 
and let 
be A-excircle .Let the two common tangents of 
cut 
in 
.Prove that 
.
55 replies
Find all integers satisfying this equation
Sadigly 2
N
by ilovenumbertheories
Source: Azerbaijan NMO 2019
Find all 
satisfying the following condition: 
satisfying the following condition: 
2 replies
Locus of the circumcenter of triangle PST
v_Enhance 14
N
by Ilikeminecraft
Source: USA TSTST 2013, Problem 4
Circle 
, centered at 
, is internally tangent to circle 
, centered at 
, at 
. Let 
and 
be variable points on and , respectively, such that line 
is tangent to (at ). Determine the locus of 
-- the circumcenter of triangle 
.
, centered at 
, is internally tangent to circle 
, centered at 
, at 
. Let 
and 
be variable points on and , respectively, such that line 
is tangent to (at ). Determine the locus of 
-- the circumcenter of triangle 
.
14 replies
Problem 6 (Second Day)
darij grinberg 43
N
by cj13609517288
Source: IMO 2004 Athens
We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers
such that has a multiple which is alternating.
Find all positive integers
such that has a multiple which is alternating.
43 replies
Dou Fang Geometry in Taiwan TST
Li4 8
N
by X.Allaberdiyev
Source: 2025 Taiwan TST Round 3 Mock P2
Let and be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and , are both on . Extend 
and 
, intersecting at and 
, respectively. Prove that 
and 
intersects on .
Proposed by kyou46, Li4, Revolilol.
and be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and , are both on . Extend 
and 
, intersecting at and 
, respectively. Prove that 
and 
intersects on .Proposed by kyou46, Li4, Revolilol.
8 replies
geometry
EeEeRUT 4
N
by Tkn
Source: Thailand MO 2025 P4
Let 
and 
be touch points of the incenter of at 
and 
, respectively. Let and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
and 
be touch points of the incenter of at 
and 
, respectively. Let and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
4 replies
min A=x+1/x+y+1/y if 2(x+y)=1+xy for x,y>0 , 2020 ISL A3 for juniors
parmenides51 14
N
by GayypowwAyly
Source: 2021 Greece JMO p1 (serves also as JBMO TST) / based on 2020 IMO ISL A3
If positive reals 
are such that 
, find the minimum value of expression 
are such that 
, find the minimum value of expression 
14 replies
Unbounded Sequences
DVDTSB 3
N
by Ciobi_
Source: Romania TST 2025 Day 2 P2
Let 
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define
![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](//latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.
Proposed by The Problem Selection Committee
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](http://latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.Proposed by The Problem Selection Committee
3 replies
Long and wacky inequality
Royal_mhyasd 1
N
by Royal_mhyasd
Source: Me
Let 
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
1 reply
Number Theory
adorefunctionalequation 3
N
by MITDragon
Find all integers k such that k(k+15) is perfect square
3 replies
