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These are answers submitted by Kitonum

It is easy to see that the product of the members of which are equidistant from the ends is 4, for example


So, we obtain

mul(simplify((sqrt(3)+tan(Pi*k/180))*(sqrt(3)+tan((30-k)*Pi/180))), k=1..14)*simplify(sqrt(3)+tan(15*Pi/180));



A:=plot3d(z(x,y), x=-4..7, y=-4..4, numpoints=3000):

x:=t->2+3*cos(t): y:=t->-1+2*sin(t):


B:=seq(plots[spacecurve]([x(t),y(t),z(x(t), y(t))], t=0..2*Pi*k/N, color=black, thickness=3), k=0..N):

bug:=seq(plottools[sphere]([x(2*Pi*k/N),y(2*Pi*k/N),z(x(2*Pi*k/N), y(2*Pi*k/N))], 0.15, style=surface, color=red),  k=0..N):

C:=seq(plots[display](A, B[k], bug[k]), k=1..N+1):

plots[display](C, insequence=true, axes=normal, view=[-4.7..7.7, -4.7..4.7, -1..5.7]);


Your task is similar to the problem of wandering drunk sailor. Do a keyword search drunken sailor.

Vector(10,[seq(sin(i), i=-0.5..-0.1, 0.1), seq(sin(i), i=0.1..0.5, 0.1)]);


Vector(10, [seq(sin(i), i in {seq(-0.5..0.5, 0.1)} minus {0.})]);

subs(a=`3`, sqrt(a)/a);



To find the center of rotation is sufficient to use two points:

restart; with(geometry):

point(X, x0, y0), point(A, 4, -2), point(B, 5, -4):   #  X - the center of rotation

L:=coordinates(rotation(A1, A, alpha, 'counterclockwise', X)):

M:=coordinates(rotation(B1, B, alpha, 'counterclockwise', X)):

solve({L[1]=4, L[2]=2, M[1]=6, M[2]=3});




You can use  seq  command.


X:=[seq(i, i=0..20)]:
Y:=[seq((-1)^i*x, i=0..19)]:
y:=piecewise(seq(op([X[i]<x and x<=X[i+1], Y[i]]), i=1..20)):
plot(y, x=0..20);




The last line should be

dsolve({eq, bcs}, v(x));

Matrix([seq([seq(a[k], k = x+m .. y+m)], m = 0 .. 2)]);

a:=1:  c:=2:

fsolve(5*b^5+(60-5*a)*b^4+(125+50*c-80*a)*b^3+(594*c-445*a-775)*b^2+(2324*c-1005*a-3270)*b+3000*c-750*a-3000=0, b);

                                          -5.950891678, -4.378215842, -3.371729567


PS. We find the real roots. If also complex roots are needed, then

fsolve(5*b^5+(60-5*a)*b^4+(125+50*c-80*a)*b^3+(594*c-445*a-775)*b^2+(2324*c-1005*a-3270)*b+3000*c-750*a-3000=0, b, complex); 

                                          -5.950891678, -4.378215842, -3.371729567,
                              1.350418543 - 1.816274549 I, 1.350418543 + 1.816274549 I

Consider the function  f:=x->sqrt(a* x + b) +  sqrt(c*x + d) . If  a*c>=0  then  f   is a monotonic function. Therefore, the equation   sqrt(a* x + b) +  sqrt(c*x + d) = m  can not have two solutions. So the necessary condition is   a*c<0 . For definiteness, let  a>0 . From these conditions and from the conditions a*x+b>=0,  c*x+d>=0  we get  -b/a<=x<=-d/c . Therefore, in the limited ranges of parameters  a, b, c, d  all the solutions can be found by the usual brute force.

The following code finds all equations with integers  a=1 .. 10,  b=-10 .. 10,  c=-10 .. -1,  d=-10 .. 10, each of which has exactly two integer solutions:


for a to 10 do

for b from -10 to 10 do

for c from -10 to -1 do

for d from -10 to 10 do

if -b/a<=-d/c then s:=floor(-b/a): t:=ceil(-d/c):


for x from s to t do

u:=sqrt(a*x+b): v:=sqrt(c*x+d):

if type(u, integer) and type(v, integer) then M:=[op(M), [x, u+v]]: fi:


if nops(M)=2 and M[1,2]=M[2,2] then L:=[op(L), [a,b,c,d,M[2,2],[M[1,1],M[2,1]]]]: fi: fi:

od: od: od: od:



ListTools[Search]([1, 5, -1, 8, 5, [-1,4]], L); 



For example, the list  [1, -9, -1, 10, 1, [9, 10]]  corresponds to the equation  sqrt(x-9)+sqrt(-x+10)=1  with the roots 9 and 10. The list  L  contains all the solutions in the specified ranges. Received 319 solutions. Displayed first 50 solutions. The original equation is also in the list  L  at position 82.

Check your syntax! I have no problem:


It makes no sense to look for what does not exist, because no global minimum and global maximum. A point  (x,y,z)  lies on the sphere of radius  sqrt(5)  and the plane  x+y+z+1=0   intersects the sphere. Therefore, the denominator of the fraction  96/(x+y+z+1)  can be arbitrarily close to 0 and may be positive or negative.

This can be seen clearly in the plot:

A:=subs(x=sqrt(5)*sin(theta)*cos(phi), y=sqrt(5)*sin(theta)*sin(phi), z=sqrt(5)*cos(theta), 1/2*x^2*y^2 + y^2*z^2 + z^2 *x^2 + 96/(x + y + z + 1)):
plot(A, theta=0..Pi, view=[0..Pi, -1000..1000], thickness=2, discont=true);



Should be


Your problem can be solved by the usual brute force. The program returns a list of all suitable  [a, b​​, c]  and the corresponding sequences  t[k], k=-1..20

t[-1]:=0: t[0]:=1: L:=[]: T:=[]:

for a to 20 do

for b to 20 do

for c to 20 do

for n from 0 to 19 do



if convert([seq(type(t[k], integer), k=1..20)], `and`) then L:=[op(L), [a,b,c]]:

T:=[op(T), [seq(t[k], k=-1..20)]] fi:

od: od: od:

L; T;


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