[/quote]
,
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
be an acute triangle and let 
be an interior point for it such that 
.
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
blocks, each weighing at least 
;
.
with 
you can choose a subset of the blocks whose total weight is at least 
.
such that 
.
chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes 
rows and 
cells in the intersection. Babek wants to have all these 
be non-negative real numbers such that 
Prove that
and 
Find the minimum 
be a positive integer and 
be real numbers such that ![\[a_1+a_2+\dots+a_n=0.\]](http://latex.artofproblemsolving.com/0/f/d/0fde8d75cebc35103ae9acf8ebe393653d336d60.png)
Define the set ![\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]](http://latex.artofproblemsolving.com/6/9/7/6974ebf20c44bcb4d52d85adc0805e28aafa5d6e.png)
Prove that, if ![\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]](http://latex.artofproblemsolving.com/b/e/3/be3e65a633dc2ac092a264968f1cf8514d3d6124.png)
is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
intersects 
distinct from 
are concyclic.
.
is a parallelogram. Prove that 
.
Y by
The circles 
and 
intersect at points 
and 
. The line 
intersects the circle at points and 
and touches the circle . The line 
intersects the circle at points and 
and touches the circle . Prove that point , point and point lie on the same line if and only if 
.
and 
intersect at points 
and 
. The line 
intersects the circle at points and 
and touches the circle . The line 
intersects the circle at points and 
and touches the circle . Prove that point , point and point lie on the same line if and only if 
.This post has been edited 1 time. Last edited by parmenides51, Apr 26, 2022, 9:37 PM
