G
Topic
First Poster
Last Poster
Stop Projecting your insecurities
naman12 53
N
by EeEeRUT
Source: 2022 USA TST #2
Let 
be an acute triangle. Let 
be the midpoint of side 
, and let 
and 
be the feet of the altitudes from 
and 
, respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point 
, and that lies on the circumcircle of . Prove that line 
is perpendicular to line .
Kevin Cong
be an acute triangle. Let 
be the midpoint of side 
, and let 
and 
be the feet of the altitudes from 
and 
, respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point 
, and that lies on the circumcircle of . Prove that line 
is perpendicular to line .Kevin Cong
53 replies
Shortest number theory you might've seen in your life
AlperenINAN 11
N
by Assassino9931
Source: Turkey JBMO TST 2025 P4
Let 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
11 replies
.
AZE JBMO TST
IstekOlympiadTeam 10
N
by Assassino9931
Source: AZE JBMO TST
Prove that there are not intgers 
and 
with conditions,
i)
is a prime number.
ii)
is a perfect square.
iii)
is also perfect square.
and 
with conditions,i)
is a prime number.ii)
is a perfect square.iii)
is also perfect square.
10 replies
Iran TST Starter
M11100111001Y1R 3
N
by dgrozev
Source: Iran TST 2025 Test 1 Problem 1
Let 
be a sequence of positive real numbers such that for every 
, we have:
![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](//latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:
![\[
a_n < \frac{1}{1404}
\]](//latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
3 replies
An interesting functional equation
giannis2006 3
N
by GreekIdiot
Source: Own
Find all functions 
such that:
for all 
The most difficult version of this problem is the following:
Find all functions such that:
for all
such that:
for all 
The most difficult version of this problem is the following:
Find all functions
such that:
for all
3 replies
1 viewing
A long non-classical problem
M11100111001Y1R 1
N
by dgrozev
Source: Iran TST 2025 Test 3 Problem 2
Suppose 
is a natural number. A function 
is called \textit{
-friendly} if for fewer than 1\% of the integers 
with 
, the equation 
has a solution in natural numbers 
such that ![\( \frac{y_0}{x_0} \in \left[\frac{1}{100}, 100\right] \)](//latex.artofproblemsolving.com/d/a/6/da67f464a934f890626d188b84940fc82c4f4ff9.png)
, where 
is a solution. Suppose 
, where 
is a polynomial with real coefficients, negative leading coefficients, and total degree greater than 2, and for every real number 
, we have 
as ![\( \frac{y}{x} \in \left[\frac{1}{100}, 100\right] \)](//latex.artofproblemsolving.com/6/5/d/65d4154dd5fb612f92f24f98406bc9bcfc713806.png)
. Prove that for sufficiently large , the function 
is not -friendly.
is a natural number. A function 
is called \textit{
-friendly} if for fewer than 1\% of the integers 
with 
, the equation 
has a solution in natural numbers 
such that ![\( \frac{y_0}{x_0} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/d/a/6/da67f464a934f890626d188b84940fc82c4f4ff9.png)
, where 
is a solution. Suppose 
, where 
is a polynomial with real coefficients, negative leading coefficients, and total degree greater than 2, and for every real number 
, we have 
as ![\( \frac{y}{x} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/6/5/d/65d4154dd5fb612f92f24f98406bc9bcfc713806.png)
. Prove that for sufficiently large , the function 
is not -friendly.
1 reply
Another FE
M11100111001Y1R 1
N
by Mathzeus1024
Source: Iran TST 2025 Test 2 Problem 3
Find all functions 
such that for all we have:
such that for all we have:
1 reply
Problem 9
SlovEcience 0
Let the sequence 
be defined by
![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](//latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)
Prove that the sequences 
and 
have finite limits, and find those limits.
be defined by![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)
Prove that the sequences 
and 
have finite limits, and find those limits.
0 replies
