,
are chosen such that when going around the triangle, the points occur in the order 
.![\[
AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2.
\]](http://latex.artofproblemsolving.com/4/c/f/4cf9e7bc009736836bcd10c1d8ba46bd3a91cd7b.png)
Prove that the perimeters of the triangles formed by the lines 
and 
are equal.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
be a cyclic quadrilateral, and let 
,
,
,
be the midpoints of 
,
,
,
respectively. Let 
,
,
and 
be the orthocenters of triangles 
,
,
and 
,
have the same area.
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...
