[/quote]
,
find the number of ordered real triples 
such that![\[\begin{aligned}
\begin{cases}
x+y^2+z^2=a,\\
x^2+y+z^2=a,\\
x^2+y^2+z=a.
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/9/8/6/986593653bf128bf7b3d497abed78e1459134c5f.png)
numbers, none of which appears more than 
times. Prove that they can be divided into 
groups of 
numbers each in such a way that the sums of the numbers in these groups are pairwise distinct.
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
such that ![\[ \dfrac{a^2}{2ab^2-b^3+1} \]](http://latex.artofproblemsolving.com/0/d/2/0d2f10786d22af2ea94ebeed6b15e723f4042734.png)
is a positive integer.
?
be a triangle and let 
and 
be variable points on sides 
and 
,
.
denote the point on 
such that 
is parallel to 
,
denote the point on 
such that 
is parallel to 
.
vary, the circumcircle of 
passes through a fixed point, other than 
.
satisfy that 
.
![\[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\]](http://latex.artofproblemsolving.com/7/9/e/79e388eba20412905964dc2b4ebb491f3c79143b.png)
and the sums ![\[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\]](http://latex.artofproblemsolving.com/3/c/4/3c44766edcc9e5269f0d3396386e41676cbdf42c.png)
For any pair define 
and 
,
.
,
such that the sequences 
and 
are unbounded while the sequence 
is bounded?
,
?
,
.
lies on 
such that 
.
passing trough 
at 
.
.
be a positive integer. Find, in terms of 
with integer coefficients such that for every integer 
such that 
.
.
Y by
Let 
be a prime number. We construct a directed graph of vertices, labeled with integers from 
to 
. There is an edge from vertex 
to vertex 
if and only if 
. Let 
denotes the length of the longest directed cycle in this graph. Prove that can attain arbitrarily large values.
be a prime number. We construct a directed graph of vertices, labeled with integers from 
to 
. There is an edge from vertex 
to vertex 
if and only if 
. Let 
denotes the length of the longest directed cycle in this graph. Prove that can attain arbitrarily large values.This post has been edited 1 time. Last edited by mathuz, Dec 28, 2023, 8:09 AM
and let 
for all 
.
,
for all positive integers 
.
;
,
,
,
,
is minimal. Then 
and we will show that 
for all positive integers 
,
doesn't have a 
term, so we have 
,
it's 
,
by induction again it's the same, so we are done.
