I want to find the numbers a, b, c, d, t, m, n of this equation. I tried
 

restart:
 k := 0:
 for a to 10 do
for b to 10 do 
for c to 10 do 
for d to 10 do 
for t to 2 do 
for m to 10 do
for n to 10 do 
if a > c and igcd(a, b, c, d, t, m, n) = 1 and abs(b)+abs(d)-n <> 0 then X := [solve(abs(a*x+b)+abs(c*x+d)-t*x^2+m*x-n = 0)]; if nops(X) = 6 and type(X[1], integer) and type(X[2], integer) and type(X[3], integer) and type(X[4], integer) and type(X[5], integer) and type(X[6], integer) then k := k+1; L[k] := [a, b, c, d, t, m, n, X[]] 
end if end if
end do end do end do end do end do end do end do; 
L := convert(L, list); 
k; 
L;

I can not get the result for along time. How can I get the result and reduce the time?

I am trying to find six integer numbers a, b, c, d, n, p so that this equation
abs(a*x+b)+abs(c*x+d)+x^2+n*x+p = 0
has 6 integer solutions are 1, 2, 3, 4, 5, 6. I tried
f:=x-> abs(a*x+b)+abs(c*x+d)+x^2+n*x+p;
solve([f(1) = 0, f(2) = 0, f(3) = 0, f(4) = 0, f(5) = 0, f(6) = 0], [a, b, c, d, n, p])

This equation has no solution. Is there six integer numbers a, b, c, d, n, p so that this equation has 6 integer solutions?

I have just found one solution is
solve(abs(-2*x+5)+abs(-2*x+9)-x^2+7*x-16 = 0, x);

With Mathematica, I see at here 
https://mathematica.stackexchange.com/questions/212808/find-integers-a-b-c-d-m-n-p-so-equation-has-six-distinct-solutions

Let be the sequence (an) so that a1 = 1, a(n+1) = 5*a(n) + sqrt(k*a(n)^2 -8), for all n >=1. I know that, the answer is k = 24.  How can I choice number k so that the sequence is an integer sequence?

I want to find coodinates of the point A, B, C, D and X of a problem 6 at IMO 2018 https://www.imo-official.org/problems.aspx

I tried 

 

restart:
 with(Student:-MultivariateCalculus):
 A := [0, 0]; 
B := [5, 0]; 
C := [3, 4]; 
DD := [a, 2]; 
solve([Distance(B, A)*Distance(C, DD) = Distance(B, C)*Distance(A, DD)], [a]);

How can I get coordinate X lies inside life request?

I want to draw circle passing thought three points S, A, B on the sphere by using tikz-3dplot-circleofsphere, its document at here https://github.com/matthias-wolff/tikz-3dplot-circleofsphere/blob/master/tikz-3dplot-circleofsphere.pdf The command to draw is 
\tdplotCsDrawCircle[style]{r}{alpha}{beta}{epsilon}

With Maple, I can find the coordinates of center and radius of circle. I tried
 

restart; 
with(geom3d):
 a := 3:
 b := 4:
 h := 5:
 point(A, 0, 0, 0):
 point(B, a, 0, 0):
 point(C, a, b, 0):
 point(DD, 0, b, 0):
 point(S, 0, 0, h):
 sphere(s, [A, B, C, S], 'centername' = m); detail(s); plane(p, [S, A, B], [x, y, z]); coordinates(projection(H, m, p)); 
R := distance(H, S)

But, I can't to find the angles alpha, beta, epsilon to draw this circle. How can I find  the angles alpha, beta, epsilon?