[/quote]
,
have the property that 
is a perfect square for infinitely many positive integers 
?
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
such that
for all real numbers 
satisfying the equation
is a prime number and 
are positive integers. Prove that there exist infinitely 
are divisible by 
be integers such that 
.
.
of positive integers, define ![\[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \]](http://latex.artofproblemsolving.com/9/4/4/9447212baebe95373b95672c72ae5b18d64fcf47.png)
Such a set 
of it, 
always holds.
.![\[ |X| \leqslant f(X) \cdot M. \]](http://latex.artofproblemsolving.com/c/7/3/c73db87bc2d9e3f44853e625625a40aa16510607.png)
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
.
and 
intersect at 
,
and 
intersect at 
and 
and 
intersect at 
.
are similar(with vertices in that order), than 
be a positive integer. There are 
real coefficient polynomials 
which is not all the same, and their leading coefficients are positive. Prove that
and find when the equality holds.
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
Y by
This could very well be reported before, however a quick search did not find anything.
While embedding LaTeX commands within BBCode (like this: [i]text $\dfrac{1}{2}$ text[/i]) typically displays as intended within a forum, trying to render a contest collection using these commands breaks the PDF renderer.
While embedding LaTeX commands within BBCode (like this: [i]text $\dfrac{1}{2}$ text[/i]) typically displays as intended within a forum, trying to render a contest collection using these commands breaks the PDF renderer.
This post has been edited 2 times. Last edited by Equinox8, Feb 26, 2025, 3:30 AM
Reason: forgot I disabled BBCode
Reason: forgot I disabled BBCode
