define 
be the product of first 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
be real numbers such that![\[a+b+c+d=-2\]](http://latex.artofproblemsolving.com/c/7/3/c73dff68dcdfbe4f6b6b2f52b378548e47193a16.png)
![\[ab+ac+ad+bc+bd+cd=0\]](http://latex.artofproblemsolving.com/9/f/4/9f450dcb150229ee76e3debdaceadddb4e451458.png)
.
and distinct positive integers 
are given. Does there exist a polynomial ![$p(x)\in\Bbb{Z}[x]$](http://latex.artofproblemsolving.com/5/1/a/51a23efd70e495d2fe516b642f0a6e71e5875ba2.png)
of degree 
that satisfies the following conditions?
of positive integers such that
with 
.
as 
,
and 
as 
.
.
and 
.
and 
.
be a point on 
such that 
.
is tangent to the circle with diameter 
and say it intersects 
.
.
as the midpoint of 
be a point on segment 
such that 
.
.
then prove 
satisfying 
Prove that 
Equality holds at 
and its cyclic permutations.
Y by
Positive integers 
satisfy both of the following conditions.
For a positive integer
, if 
, then 
.
There exist integers
that satisfies the equation 
and 
.
Prove that there exist integers that satisfies the equation 
, for each integer 
.
satisfy both of the following conditions.For a positive integer
, if 
, then 
.There exist integers
that satisfies the equation 
and 
.Prove that there exist integers
that satisfies the equation 
, for each integer 
.This post has been edited 4 times. Last edited by Hip1zzzil, Mar 30, 2025, 1:07 PM
Reason: Better
Reason: Better
be a solution of 
,
,
,
with 
gives all 
,
and 
are square free and coprime.
so that 
.![\[ (\zeta t + p)^2 + (\omega t +q)^2 -a(\alpha t +r)^2 -b(\beta t +s)^2 = 2(\zeta p + \omega q -a\alpha r -b \beta s)t + p^2 +q^2 -ar^2 -bs^2 \]](http://latex.artofproblemsolving.com/9/c/c/9ccadb23a0effe0bce5929a9ee9297a88f46e93a.png)
From Bezout, we have 
such that![\[ \zeta p + \omega q -a\alpha r -b \beta s =1 \]](http://latex.artofproblemsolving.com/e/d/5/ed551a451a3e968217b90a7aee89058b1d687ff3.png)
so we have all ever, or all odd 
,
,
are all odd, so for that case I used ![\[ \zeta p + \omega q -a\alpha r -b \beta s =2 \]](http://latex.artofproblemsolving.com/0/c/e/0ce66f661dbb7eb85a13ba230006a2c72b67cdde.png)
and eventually I got all odds, 
s
s
is not squarefree then 
is a positive integer...
