PROBLEMS Vol. 102(2), February 2002

•  4713: Proposed by: Michael Brozinsky, Central Islip, NY

According to legend, the only way that a famous pirateâs chest could be opened was to put keys numbered from 1 to m (m Ò 2) in numerical order into the lock while keeping all used and unused keys together in a jar to prevent any of the keys from getting lost. Due to corrosion, the numbers on the keys had become obliterated, and the keys themselves could in no way be marked to distinguish one from another (that is, we can no longer decide whether a given key has been tried or not). What is the expected number of keys that will be tried before the chest is opened? (Equivalently, if an m sided die labeled with the positive integers 1,2,3,. ..,m is flipped repeatedly, what is the expected number of flips until the sequence of consecutive integers 1,2,3,4,.. .m results? 

ð    4714: Proposed by: Richard L. Francis, Cape Girardeau, MO.

The trisectors of the angles of any triangle intersect in the vertices of an equilateral triangle. This equilateral triangle is called the Morley triangle of the given triangle. Is the ratio of the perimeter of a triangle to the perimeter of its Morley triangle the same for all triangles? 

ð    4715: Prove that there is no circle with center (√2, √3) that has two or more lattice points on its circumference. (A point (m, n) in the plane is a lattice point if each of its coordinates is an integer.) 

ð    4716: Proposed by: Richard L. Francis, Cape Girardeau, MO.

It is well known that the constructible 60¡ angle is not trisectible by Euclidean methods. Show that infinitely many smaller constructible angles other than those of the form 60¡/2ⁿ are also non-trisectible. 

ð    4717: Show that for all integers n, the number 8,640 divides n⁹- 6n⁷ + 9n⁵- 4n³. 

ð    4718: If x, y, and z are selected independently and at random from the interval [0, 1], what is the probability that xÒ Ò yz?