[/quote]
,
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
with real coefficients of degree 
such that 
divides 
.
such that:
.
.
so that any 
.
satisfying the following:
;
and 
;
;
has all of its coefficients being even.
and 
Prove that
points on the plane with diameter 
.
and two not necessarily distinct points 
not equal to 
or 
satisfying that![\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]](http://latex.artofproblemsolving.com/f/3/6/f3649872c33e1db8880b4aedace38d9439f2b4ae.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
of positive integers for which there exist positive integers 
and 
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
such that 
for all 
?
and 
Prove that
Y by Fishheadtailbody
Given 
students and their test results on 
different subjects, we say that student 
dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.
Well, it seems like more clarity and formality are needed.
Given vectors in a -dimensional space, 
, 
. We say that a vector 
is dominant to another vector 
if and only if all components of are greater than those of . Assume that no components are the same, that is 
.
Find the probability that there exists a pair of vectors that one is dominant to another.
students and their test results on 
different subjects, we say that student 
dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.Well, it seems like more clarity and formality are needed.
Given
vectors in a -dimensional space, 
, 
. We say that a vector 
is dominant to another vector 
if and only if all components of are greater than those of . Assume that no components are the same, that is 
.Find the probability that there exists a pair of vectors that one is dominant to another.
This post has been edited 1 time. Last edited by Entei, Apr 6, 2025, 5:48 PM
Reason: Clarity and formality.
Reason: Clarity and formality.
