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be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .
, then either travels [list] units vertically (up or down) and 
units horizontally (left or right); or units horizontally (left or right) and units vertically (up or down)., the ant can choose to go to one of eight possible points.
and 
, the ant can travel from 
to 
using at most 
moves.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both and . Integers and are coprime if 
.
, the external bisectors of 
and 
meet at a point 
. Prove that the circumcentre of 
and the points 
lie on the same straight line.
. How many complex numbers 
are there such that 
and 
is an integer.
be reals such that 
. Prove that
Let be reals such that 
. Prove that
and 
for each positive integer 
, what is the value of 
?
be a positive integer such that 
, and let the set![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
, with 
, if 
, then 
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
for which I can find nonnegative integers 
such that for all indices 
, 
divides 
?
...") for which I there exists nonnegative integers such that for all indices , divides ?
integers - take the set of all integers with a digit sum of 
in base , then append 
to the list.
be a non-isosceles triangle with orthocenter 
, circumcenter 
. Let 
be midpoints of 
and 
be reflections of across 
. Prove that triangle 
have the same centroid.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
and 
Prove that
be real numbers such that 
Prove that
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
