Another right angled triangle
by ariopro1387, May 25, 2025, 4:13 PM
Let 
be a right angled triangle with 
. Point 
is the midpoint of side 
And 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
be a right angled triangle with 
. Point 
is the midpoint of side 
And 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
such that 
My Unsolved Problem
by ZeltaQN2008, May 24, 2025, 10:18 AM
Given a positive integer 
and 
. Prove that there always exists a positive integer 
such that 
.
P/s: I can prove the problem if
is a power of a prime number, but for arbitrary then well.....
and 
. Prove that there always exists a positive integer 
such that 
.P/s: I can prove the problem if
is a power of a prime number, but for arbitrary then well.....Problem 7
by SlovEcience, May 14, 2025, 11:03 AM
Consider the sequence 
defined by 
and
![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](//latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.
b) Compute
![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](//latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.b) Compute
![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
ISI UGB 2025 P2
by SomeonecoolLovesMaths, May 11, 2025, 11:16 AM
If the interior angles of a triangle satisfy the equality, 
prove that the triangle must have a right angle.
satisfy the equality, 
prove that the triangle must have a right angle.This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, May 11, 2025, 12:00 PM
Functional Inequaility
by ariopro1387, Apr 9, 2025, 2:39 PM
Find all functions 
such that for any real numbers 
and 
, the following inequality holds:
![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](//latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
such that for any real numbers 
and 
, the following inequality holds:![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
Lots of perpendiculars; compute HQ/HR
by MellowMelon, Jul 26, 2011, 9:14 PM
In an acute scalene triangle , points 
lie on sides 
, respectively, such that 
. Altitudes 
meet at orthocenter 
. Points and lie on segment 
such that 
and 
. Lines 
and 
intersect at point 
. Compute 
.
Proposed by Zuming Feng
, points 
lie on sides 
, respectively, such that 
. Altitudes 
meet at orthocenter 
. Points and lie on segment 
such that 
and 
. Lines 
and 
intersect at point 
. Compute 
.Proposed by Zuming Feng
sequence (.) eventually becomes constant.
by N.T.TUAN, Apr 26, 2007, 5:54 AM
Let 
be a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.
be a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.♪ i just hope you understand / sometimes the clothes do not make the man ♫ // https://beta.vero.site/
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