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Stop Projecting your insecurities
naman12 53
N
12 minutes ago
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














Kevin Cong
53 replies

Shortest number theory you might've seen in your life
AlperenINAN 11
N
18 minutes ago
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.




11 replies


AZE JBMO TST
IstekOlympiadTeam 10
N
30 minutes ago
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.


i)

ii)

iii)

10 replies

Iran TST Starter
M11100111001Y1R 3
N
32 minutes ago
by dgrozev
Source: Iran TST 2025 Test 1 Problem 1
Let
be a sequence of positive real numbers such that for every
, we have:
Prove that there exists a natural number
such that for all
, the following holds:


![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)


![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
3 replies
An interesting functional equation
giannis2006 3
N
33 minutes ago
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



The most difficult version of this problem is the following:
Find all functions



3 replies
1 viewing
A long non-classical problem
M11100111001Y1R 1
N
36 minutes ago
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
, 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
. Prove that for sufficiently large
, the function
is not
-friendly.







![\( \frac{y_0}{x_0} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/d/a/6/da67f464a934f890626d188b84940fc82c4f4ff9.png)





![\( \frac{y}{x} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/6/5/d/65d4154dd5fb612f92f24f98406bc9bcfc713806.png)



1 reply
Another FE
M11100111001Y1R 1
N
an hour ago
by Mathzeus1024
Source: Iran TST 2025 Test 2 Problem 3
Find all functions
such that for all
we have:



1 reply


Problem 9
SlovEcience 0
an hour ago
Let the sequence
be defined by
Prove that the sequences
and
have finite limits, and find those limits.

![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)


0 replies
