I am trying to start a weekly math contest at my school. I have enjoyed the task of finding problems to use. This contest finishes today for my students, so I will pass on the questions for your own enjoyment. There is no prize but the dignity and warmth that comes of mathematical triumph. I will post answers in the comments when I have the time and inclination.
Grades 11 & 12 5-Star Challenge #1
Mr. Burchell has two identical vases and he wants to know exactly how far such a vase could fall without breaking. The vases may be fairly robust. With no more than 13 drops from integer heights (in feet), what is the greatest height (in feet) which could be determined with certainty as the longest drop which such a vase could survive (without breaking)? That is, beginning at 1 foot, what is the greatest number of heights which could be proven safe for the vase?
Grades 9 & 10 5-Star Challenge #1
Circle P with radius 9 and circle Q with radius 16 intersect only once, at point A. A certain line is tangent to both circles, touching circle P at point B and circle Q at point C. Find the area of triangle ABC.