minhthien2016

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7 years, 50 days

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These are questions asked by minhthien2016

I am trying to find the number a and b (-20 < a < 20, -20 < b <20) so that two circles (x+1)^2+(y+3)^2 = 125 and (x-a)^2+(y-b)^2 = 225 cut at two points A and B and coordinates A and B are pairs of integers.
I tried
 

restart; L := []; 
for a from -20 to 20 do 
for b from -20 to 20 do 
for x from -20 to 20 do 
for y from -20 to 20 do 
if (x+1)^2+(y+3)^2 = 125 and (x-a)^2+(y-b)^2 = 225 and nops({a, b, x, y}) = 4 and x*y*a*b <> 0 then 
L := {op(L), {[a, b], [x, y]}} 
od: od: od: od:
nops(L); 
L

How to select the number a and b so that the system of equations (x+1)^2+(y+3)^2 = 125  and (x-a)^2+(y-b)^2 = 225 have two integral solutions. For example
 

restart; 
solve({(x-6)^2+(y+2)^2 = 225, (x+1)^2+(y+3)^2 = 125}, {x, y})


I want to find general formulas of the sequence satifies:
f(1) = 1, f(n+1) = 1/2*(f(n)+9/f(n))
I tried 
restart;
rsolve({f(1) = 1, f(n+1) = 1/2*(f(n)+9/f(n))}, {f})

I got the answer
3*coth((1/2)*arccoth(1/3)*2^n)

How to get the form (-3*2^(2^(-1+n))+3*(-1)^(1+2^(-1+n)))/(-2^(2^(-1+n))+1)?

I am trying to find the positive integer numbers a, b, c, d, e, f so that the function y =  (a x^2 + b x + c)/(d x^2 + e x + f) is increasing function and the equation  (a x^2 + b x + c)/(d x^2 + e x + f) = x has three integer solutions. I tried
restart;
ListTools[Categorize];
L := [];
for a to 10 do
for b to 10 do
for c to 10 do
for d to 10 do
for e to 10 do
for f to 10 do
if a*e-b*d > 0 and a^2*f^2-a*b*e*f-2*a*c*d*f+a*c*e^2+b^2*d*f-b*c*d*e+c^2*d^2 < 0 and igcd(a, b, c) = 1 and igcd(d, e, f) = 1 and igcd(a, b, c, d, e, f) = 1 and x = (a*x^2+b*x+c)/(d*x^2+e*x+f) then L := [op(L), [a, b, c, d, e, f, x]]

L:=[op(L), [a, b, c, d,e,f,x]]; fi; 
od: od: od: od:od:od: 
nops(L);  


I did not get the result. 
With Mathematica, I got like this picture (a, b, c, d, e, f, x1, x2, x3)

How can I get the correct results?

I am trying to find formula of the sequence f(n)

f(n+1) = (1+f(n))/(2*f(n)), f(1) = -7/13

I tried
restart;
rsolve({f(1) = -7/13, f(n+1) = (1+f(n))/(2*f(n))}, {f})


But I couldn't get the result. With Mathematica, I got the result.

I want to find all complex numbers such that 
abs(z)*(z-4-I)+2*I = (5-I)*z.
I tried solve(abs(z)*(z-4-I)+2*I = (5-I)*z, z)
and got the answer is z = -1.
This is a question in a test with multiple choice. The key of question is three numbers.

Is my commant wrong? Where is wrong?

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