A regular octahedron is made up of eight equilateral triangles, each with side length one unit, as shown below. An ant starts at the top vertex, walks along the edges of the triangles without ever traversing the same edge twice, and ends at the top vertex. If she did not pass through the top vertex at any other point in her walk, how many units are in the maximum distance she could walk?

2. Let a,b c,d, and e be positive integers. The sum of the four numbers on each of the five segments connecting "points" of the star is 28. What is the value of the sum a+b+c+d+e?

3. The product of a list of (not necessarily distinct) positive integers is 120. What is the least possible sum of the numbers in the list?

Valencia Aug 10, 2021