kids in a class. Is it true that for all positive integers 
has a right angle at 
.
,
is called 
.
lies on 
with 
.
meets 
in 
.
and 
intersect in 
.
and 
are parallel.
let 
denote the number of (positive) divisors of 
.
with the following properties: [list][*] 
for all 
.
divides 
for all 
,
.
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
![\[\sum_{n=1}^{2025}n^5\]](http://latex.artofproblemsolving.com/0/8/3/08334297eaac7610645549ede6434e399000e480.png)
numbers can be expressed in 
digits in base 
digits in base 
,
for some positive integers 
.
satisfies the following conditions:![\begin{align*}
&f(4xy)=2y[f(x+y)+f(x-y)]\text{ for all }x,y\in\mathbb R\\
&f(5)=3
\end{align*}](http://latex.artofproblemsolving.com/6/2/5/625b8046a00d07680dd2a8d2aa3231d5daa4fcf2.png)
.
is an ordered sequence of positive integers greater than one satisfying the following conditions:
,
.
is a composite number, then there exists at least 1 positive integer 
,
,
.
terms.
?
denote the sum of the digits of 
.
,
and 
?
,
are positive integers?
,
(the "discriminant" is exactly 
more than the first term, and the fourth term is 
more than the second term. Find the first term.
such that 
.
Y by
Let 
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
