[/quote]
,
of acute triangle 
is 
.
intersects 
at 
.
be 
.
,
,
,
,
,
similarly. Prove that 
,
,
are concurrent.
to 
(including 
(multiplication) between each number. Then he writes the rest of the numbers in the denominator and also writes 
.![\[\sqrt{a^2+1}+\sqrt{b^2+1}+\sqrt{c^2+1}\le \sqrt{2}(a+b+c)\]](http://latex.artofproblemsolving.com/6/e/d/6ed654c1d3616a8274ff0838e7bb13dfa0a11b8a.png)
light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number 
for which we can surely turn on at least 
denote the infinite parallel ruler with the parallel edges being at distance 
from each other. The following construction steps are allowed using ruler 
:
parallel to a given line 
a
with 
two parallel lines at distance 
,
be a given positive integer. Prove that based on the three points and 
such that for any 
it is possible to construct 
and 
is non-decelerating if 
holds for all positive integers 
is convergence-inducing, if the following statement is true for all real sequences 
:
is convergent and tends to 
for all positive integers 
,
,
,
is bounded from above.
light bulbs are arranged around a circle, and are consecutively numbered with
.
,
and 
denote the feet of the altitudes from 
and 
,
intersect circumcircle 
.
intersect circumcircle 
.
and 
passes through the orthocenter of triangle 
,
,![\[
a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},
\]](http://latex.artofproblemsolving.com/4/e/c/4ec892df19b4ec97af9bb1e5d1954e6c26be3172.png)
for all integers 
.
such that![\[
\frac{a_n}{n}
\]](http://latex.artofproblemsolving.com/3/d/1/3d1d012e7c41e1e09abd14626a79b1fb1cbab34d.png)
is an integer.
is clearly not true. Even a 1st grader can see that. And what does "a number not a product" mean? A product is a number? Literally makes 0 sense.
.
as 
?
if you do go around, acting like some kind of god again, "anticodon." Everyone will worship you finally for being able to copy from Google correctly!...
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
