Maximum number of sides of a convex polygon Exactly 3 of the interior angles of a convex polygon are obtuse. What is the maximum number of sides? 
 A: We will show that the maximum number of sides is 6.
The sum of the interior angles of a convex polygon of $n$ sides is $(n-2)180^{\circ}$. Hence the sum of the exterior angles is $n180^{\circ}-(n-2)180^{\circ}=360^{\circ}$. It follows that we can not have 4 or more obtuse exterior angles that is 4 or more acute interior angles.
Moreover, note there is a convex hexagon with angles:
$170^{\circ},170^{\circ},170^{\circ},70^{\circ},70^{\circ},70^{\circ}$.
Just glue three isosceles triangles with angles $170^{\circ},5^{\circ},5^{\circ}$ around an equilateral triangle (with side equal to the base of the isosceles triangles).
A: Hint 1:  If an interior angle is acute, the exterior angle is obtuse, and vice versa.
Hint 2:  What do the exterior angles of a convex polygon sum to?
Added later:  The answer to the OP's question can be $6$, $7$, or $\infty$, depending on the precise definitions of "convex" and "obtuse."
If the interior angles of a convex polygon are required to be strictly less than $180$ degrees, then the answer is $6$, as Robert Z's answer shows.
If interior angles of $180$ degrees are allowed, and these are considered obtuse (because $180\gt90$), then the answer is $7$:  Take, for example, a square, with $4$ right angles and designate any $3$ points along any of the sides of the square as additional vertices, each with an obtuse angle of $180$ degrees.  By Robert Z's argument (and/or the hints above), you can't do better than $7$.
Finally, if interior angles of $180$ degrees are allowed, but obtuse angles are required to be strictly between $90$ and $180$ degrees (which seems to be a standard definition of "obtuse"), then you can take any convex polygon with three obtuse angles and designate as many more points as you please along the perimeter as additional vertices, each with a (non-obtuse!) interior angle of $180$ degrees.  In this case, there is no upper bound on the number of sides, so we can say the answer is $\infty$.
In comments, the OP indicated that convexity does allow for interior angles of $180$ degrees but did not specify whether they are considered obtuse.  In either case, the answer $6$ is technically wrong for this definition.
