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Stop Projecting your insecurities
naman12   53
N 12 minutes ago by EeEeRUT
Source: 2022 USA TST #2
Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$.

Kevin Cong
53 replies
naman12
Dec 12, 2022
EeEeRUT
12 minutes ago
Roots of unity
Henryfamz   1
N 18 minutes ago by Mathzeus1024
Compute $$\sec^4\frac\pi7+\sec^4\frac{2\pi}7+\sec^4\frac{3\pi}7$$
1 reply
Henryfamz
May 13, 2025
Mathzeus1024
18 minutes ago
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 $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.
11 replies
AlperenINAN
May 11, 2025
Assassino9931
18 minutes ago
Inspired by qrxz17
sqing   1
N 23 minutes ago by lbh_qys
Source: Own
Let $a, b,c>0 ,(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) = 27 $. Prove that $$a+b+c\geq 3\sqrt {3}$$
1 reply
sqing
an hour ago
lbh_qys
23 minutes ago
AZE JBMO TST
IstekOlympiadTeam   10
N 30 minutes ago by Assassino9931
Source: AZE JBMO TST
Prove that there are not intgers $a$ and $b$ with conditions,
i) $16a-9b$ is a prime number.
ii) $ab$ is a perfect square.
iii) $a+b$ is also perfect square.
10 replies
IstekOlympiadTeam
May 2, 2015
Assassino9931
30 minutes ago
Iran TST Starter
M11100111001Y1R   3
N 32 minutes ago by dgrozev
Source: Iran TST 2025 Test 1 Problem 1
Let \( a_n \) be a sequence of positive real numbers such that for every \( n > 2025 \), we have:
\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]Prove that there exists a natural number \( M \) such that for all \( n > M \), the following holds:
\[
a_n < \frac{1}{1404}
\]
3 replies
M11100111001Y1R
May 27, 2025
dgrozev
32 minutes ago
An interesting functional equation
giannis2006   3
N 33 minutes ago by GreekIdiot
Source: Own
Find all functions $f:R^+->R^+$ such that:
$f(xf(y))=xy-xf(x)+f(x)^2$ for all $x,y>0$

The most difficult version of this problem is the following:
Find all functions $f:R^+->R^+$ such that:
$f(xf(y+f(x)))=xy+f(x)^2$ for all $x,y>0$
3 replies
1 viewing
giannis2006
Jun 8, 2023
GreekIdiot
33 minutes ago
A long non-classical problem
M11100111001Y1R   1
N 36 minutes ago by dgrozev
Source: Iran TST 2025 Test 3 Problem 2
Suppose \( n \in \mathbb{N} \) is a natural number. A function \( f(x, y) \) is called \textit{\( n \)-friendly} if for fewer than 1\% of the integers \( k \) with \( -n \leq k \leq n \), the equation \( f(x, y) = k \) has a solution in natural numbers \( (x, y) \) such that \( \frac{y_0}{x_0} \in \left[\frac{1}{100}, 100\right] \), where \( (x_0, y_0) \) is a solution. Suppose \( f(x, y) \leq g(x, y) \), where \( g(x, y) \) is a polynomial with real coefficients, negative leading coefficients, and total degree greater than 2, and for every real number \( x \), we have \( g(x, y) \to \infty \) as \( \frac{y}{x} \in \left[\frac{1}{100}, 100\right] \). Prove that for sufficiently large \( n \), the function \( f \) is not \( n \)-friendly.
1 reply
M11100111001Y1R
May 27, 2025
dgrozev
36 minutes ago
Another FE
M11100111001Y1R   1
N an hour ago by Mathzeus1024
Source: Iran TST 2025 Test 2 Problem 3
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x,y>0$ we have:
$$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$$
1 reply
M11100111001Y1R
2 hours ago
Mathzeus1024
an hour ago
Problem 9
SlovEcience   0
an hour ago
Let the sequence $(x_n)$ be defined by
\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]Prove that the sequences \( y_n = \frac{x_n}{n} \) and \( z_n = x_n - n \) have finite limits, and find those limits.
0 replies
SlovEcience
an hour ago
0 replies
a