[/quote]
,
,
,
are real numbers that satisfy:![\[ x=\sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}} \]](http://latex.artofproblemsolving.com/e/a/5/ea51d0d39dd7182d4145bb90349c1c389dec7ee1.png)
![\[ y=\sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}} \]](http://latex.artofproblemsolving.com/2/3/0/23017c3c8334dcb3c0492a662a9b75408e8cd455.png)
![\[ z=\sqrt{x^2-\frac{1}{36}}+\sqrt{y^2-\frac{1}{36}} \]](http://latex.artofproblemsolving.com/f/1/6/f1621872c6453052422074bad4d9bb12c1fb33e9.png)
,
and 
are positive integers and 
.
be a triangle with orthocentre 
.
be the foot of altitudes of 
onto the opposite sides, respectively. Consider 
,
about line 
.
cut 
,
be the orthocenter of 
.
are concyclic.
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
.
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
such that 
is chosen on the side 
of a triangle 
in such a way that 
.
and 
and 
,
be the incenter of triangle 
.
intersects the line segment 
at its midpoint.
be an integer. We call a pair of integers 
g![\[0\leqslant a<k,\hspace{0.2cm} 0<b \hspace{1cm} \text{and} \hspace{1cm} (a+b)^2=ka+b\]](http://latex.artofproblemsolving.com/e/8/d/e8df562e024b8476d69fdaea15338c27bb6c4eb8.png)
Prove that the number of 
.
be an equilateral triangle with incircle 
.
.
![\[
\frac{3 + \sqrt{4n + 9}}{2}
\]](http://latex.artofproblemsolving.com/f/2/f/f2f09c19d46025866c5116d6909b3e9ab5e2ac66.png)
is the sixth smallest positive divisor of 
be real numbers such that![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](http://latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
.
satisfying 
and![\[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.\]](http://latex.artofproblemsolving.com/0/4/4/0446861fb2e63782803c4f182a529b79e213fdfd.png)
![\[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\]](http://latex.artofproblemsolving.com/e/8/7/e87fdc9fe63d29a46e57dfb92c2d0448441b4b55.png)
There are positive integers 
such that 
.
.
Y by
I got an email from HCSSiM titled “Yes, please soon, HCSSiM '24, [insert name] afm” and it starts with “Your personal “Yes, please” invitation to be a part of the 2024 Hampshire College Summer Studies in Mathematics is forthcoming.”
Then it asked me for more thoughts on parts of the IT that I got wrong on. The email was very warm and genuine. I heard that an acceptance letter has the title “Yes, please [insert name]”, so I wasn’t sure if my email means an acceptance also.
Did anyone else get something similar to this?
Then it asked me for more thoughts on parts of the IT that I got wrong on. The email was very warm and genuine. I heard that an acceptance letter has the title “Yes, please [insert name]”, so I wasn’t sure if my email means an acceptance also.
Did anyone else get something similar to this?
