AoPS Academy
kids in a class. Is it true that for all positive integers 
it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
has a right angle at 
. The centre of the circumcircle is called 
, and the base point of the normal from to 
is called 
. The point 
lies on 
with 
. The angle bisector of 
meets 
in 
. The lines 
and 
intersect in 
. Show that the lines 
and 
are parallel.
let 
denote the number of (positive) divisors of 
. Find all functions 
with the following properties: [list][*] 
for all 
.
divides 
for all 
, 
.[/list]
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
numbers can be expressed in 
digits in base , 
digits in base 
, and 
digits in base 
for some positive integers 
. How many such two-digit base numbers are there?
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:
, then 
.
is a composite number, then there exists at least 1 positive integer 
, 
, such that 
.
terms.
?, let 
denote the sum of the digits of . Determine the value of 
.
, where 
and are relatively prime positive integers. What is 
?
, given that 
are positive integers? that make this possible?
, the difference is 
(the "discriminant" is exactly greater than the area).
more than the first term, and the fourth term is 
more than the second term. Find the first term.
such that 
.
![\[\sum_{n=1}^{2025}n^5\]](http://latex.artofproblemsolving.com/0/8/3/08334297eaac7610645549ede6434e399000e480.png)
has side lengths 
,
,
.
inside 
,
,
and denote them as 
,
,
.
?
