by reni_wee, Apr 29, 2025, 9:45 AM 
be a triangle with orthic triangle 
and orthocenter 
and midpoint of 
as 
. If 
then prove that 
passes through pole of wrt 
.by Mathandski, Apr 28, 2025, 7:58 PM 
such that 
and 
, evaluate 
.by flower417477, Apr 28, 2025, 3:58 PM with incenter 
,
is a point on 
,the bisector of 
meet 
at 
.Prove that 
by CarSa, Apr 27, 2025, 1:18 AM 
to the system of equations![\[\begin{aligned}
\begin{cases}
x^2+y^2-xy-3x+3=0,\\
x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/4/a/b/4ab8e94e2e39be219c80264aa5ea23456b74b3e8.png)
by KhuongTrang, Apr 24, 2025, 2:52 PM 
Prove that
by KhuongTrang, Apr 21, 2025, 4:18 PM 
by mshtand1, Apr 19, 2025, 9:31 PM 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
by egxa, Apr 18, 2025, 5:12 PM 
sports, each awarding one gold medal to a winner. athletes participate, each in all sports. There are also experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to . An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number 
such that the experts can make their predictions so that at least of them are guaranteed to be competent regardless of the outcome?by MellowMelon, Jul 26, 2010, 4:03 PM 
by 
and, for 
,![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](http://latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
such that 
.by joml88, Dec 9, 2005, 12:12 AM 
has exactly two real roots, one of which is 
where 
and 
are integers, 
and are relatively prime, and 
Find 
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