[/quote]
,
satisfy the equality, 
prove that the triangle must have a right angle.
be a cyclic quadrilateral and let the intersection point of lines 
and 
be 
.
and 
be arbitrary points on sides 
Prove that 
.
square is filled with numbers 
.
squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
satisfying 
,
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
so that for any 
,![\[ f(xf(x^2)+yf(x))=xf(y+xf(x)). \]](http://latex.artofproblemsolving.com/1/d/4/1d4fbf3f80306aaae5cfc11cc0e4be7ea41491d7.png)
such that a positive integer power of 
equals to a positive integer power of 
.
positive real numbers satisfying 
,![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
data values 
are all equal to 
.
plot 
,
distinct moves available; two from each point. Let 
count the number of ways to advance around the circle exactly twice, beginning and ending at 
for all 
.
be the real solution of the system of equations 
,
,
.
can be written in the form 
,
and 
are relatively prime positive integers. Find 
.
Y by Danielzh, BS2012, EpicBird08, bachkieu
I got an email about new FedEx shipment earlier today. I never ordered anything and was pretty confused but it caught my interest because it shipped out of Elgin, IL, which is only ~15 miles from the place where MOP is taking place and was shipped directly to my name and the email I signed up to AMCs with (which I don't use for much other things).
This is a very stupid question and it might be a coincidence but did any other AoPSers waiting on MOP email receive this ;-;
This is a very stupid question and it might be a coincidence but did any other AoPSers waiting on MOP email receive this ;-;
This post has been edited 1 time. Last edited by Mathandski, Apr 18, 2025, 9:00 PM
