## Puzzle of the week [15]

**T**he puzzle in the last weekend edition of * Le Monde* is simple to state: given a rectangle with sides

*1*and

*a<1*, what is the condition on a for an equilateral triangle to be inscribed in it? (Meaning that one summit of the triangle must coincide with one summit of the rectangle and that both other summits must stand on opposite sides of the rectangle.)

**T**he equality on the sides of the triangle leads to finding solutions in *(b,c)* to the equations

Eliminating *c* leads to an equation in *b* of the form

which can only have a solution for . If we rewrite and , this equation turns into

with the constraint that . Since the right hand side is decreasing in v, the equation has a solution if the rhs takes different signs for and . This is only the case when , which is equivalent to

.

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