minhthien2016

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7 years, 131 days

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These are questions asked by minhthien2016

How do I draw this polyhedron knowing all vertices?
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_21?fbclid=IwAR2bdHwJEpjarZEwope1hYOcnnXqnvzae38Y8ZrfWKCKQunfAuT9s30q32o

[[-1.5, -0.5, 0.], [-1.5, 0.5, 0.], [-1., -1., -0.707107], [-1., 1., -0.707107], [-0.5, -1.5, 0.], [-0.5, -0.5, -1.41421], [-0.5, 0.5, -1.41421], [-0.5, 1.5, 0.], [0.5, -1.5, 0.], [0.5, -0.5, -1.41421], [0.5, 0.5, -1.41421], [0.5, 1.5, 0.], [1., -1., -0.707107], [1., 1., -0.707107], [1.5, -0.5, 0.], [1.5, 0.5, 0.]]

 

The date, month, year  22/02/2022 is said a beautiful numbers. Because, the number 2202 symmetry to the number 2022. How can I find another beautiful numbers like that ?

With two vectors a and b, we know that
Norm(CrossProduct(a, b)) = Norm(a)* Norm(b) * sin(a,b).
I tried with a := <1, 2, -2>; b := <2, 10, 11>; 

Note that a perpendicular to b and 

Norm(CrossProduct(a, b)) = Norm(a)* Norm(b)

I tried

restart;
with(VectorCalculus);
SetCoordinates(cartesian[x, y, z]);
a := <1, 2, -2>;
b := <2, 10, 11>;
Norm(a);
Norm(b);
v := CrossProduct(a, b);
Norm(v);


Are there two vectors a and b with integer coordinates and  not perpendicular,  so that Norm(a), Norm(b), Norm(CrossProduct(a, b)) are interger numbers satisfying
Norm(CrossProduct(a, b)) = Norm(a)* Norm(b) * sin(a,b).

The system of equations x*y*z + y*z + y = 21, x*y*z + x*z + z = 30, x*y*z + x*y + x = 12

has three solutions, one of them is not an integer solution.

solve({x*y*z + y*z + y = 21, x*y*z + x*z + z = 30, x*y*z + x*y + x = 12}, {x, y, z})

How can I find three numbers a, b, c so that the system of equations

x*y*z + y*z + y = a, x*y*z + x*z + z = b, x*y*z + x*y + x = c

has three  solutions (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), where x1, y1, z1, x2, y2, z2, x3, y3, z3 are nine integer numbers.

Tetrahedron with length of sides like this picture has volume is an integer number. Is there another tetrahedron like that?

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