Summary: Some problems from the American Mathematical Monthly
1. The perimeter of a triangle ABC is divided into three equal parts by three points P, Q, R. Show that
and that the constant 2/9 is the best possible.
2. Find all pairs of positive integers m, n such that (m)|n and (n)|m, where denotes Euler's function.
3. Let S be the boundary of the unit square [0, 1] × [0, 1] in R2
. Suppose f is a continuous real-valued
function on S such that f(x, 0) and f(x, 1) are polynomial functions of x on [0, 1] and such that f(0, y)
and f(1, y) are polynomial functions of y on [0, 1]. Prove that f is the restriction to S of a polynomial
function of x and y.
4. Suppose n points are independently chosen at random on the perimeter of a circle. What is the proba-
bility that all the points lie in some semicircle?
5. A population consisting of particles of various types evolves in time according to the following rule: Each
particle is deemed to belong to a unique generation n = 1, 2, 3, . . . . Each particle produces a certain
number of "offspring" particles, and, for each n, generation n + 1 comprises the totality of offspring of
the particles in generation n. A particle of type i = 0, 1, 2, . . . produces exactly i + 2 offspring, one each
of types 0, 1, 2, . . . , i + 1. Let N(n, k) denote the number of particles in the nth generation when the