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.

I think the greatest height is 338, but now I want to know if I am correct. 13 drops from each vase gets you a total of 26 drops. You start by dropping at 26 feet if it breaks just work your way down to get the highest. height if it does not break you have 25 more drops so add 25 to the orginal 26. Continue this pattern you get 338.

ReplyDeleteI intended that there would be 13 drops total, but that should maybe be more clear. If there are 13 drops for each vase, you could not start at 26 because if it broke, you would only have 13 drops left on the remaining vase.

ReplyDeleteMy answer is 25 feet. Drop the first vase from these heights: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 25. If at any point the first vase breaks, drop the second vase from one foot lower.

ReplyDeleteOf course, this whole problem assumes that the vases are not weakened by repeated successful drops...

ReplyDelete...and it further assumes that a successful drop establishes the safety of lesser drops.

ReplyDeleteThe vase question : I can't help feeling that there is a twist to the question. The vases would survive any one drop from any height until the last foot( for instance)- whence upon hitting a rigid surface they would shatter.

ReplyDeleteThe circles question : The area of the triangle should be zero since all points coincide - the tangents and the point of intersection of the circles.