<s>[INST] The top of a tree is broken off in a storm, and the top part falls to the ground but does not get uprooted. Find the ratio of the height of the stump to the height of the tree before the break. This ratio does not depend on the height of the tree!
Let the break be at a height h above the ground. Let the height of the broken part be x. Obviously, the break occurs at a height above the ground equal to the height of the broken part above the ground when it is laid along the ground.
From the triangle similarity we get:

h/x = (h + x)/2x

Simplifying the equation we get:

2hx = hx + x^2
x^2 + hx - 2hx = 0
x^2 - hx = 0

Divide both sides by x we get:

x - h = 0
x = h

Now we can calculate the desired ratio:

(H - h)/H = 1 - h/H = 1 - 1/2 = 1/2

Wait, what?[/INST]