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 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 Area(PQR) > 2 9 Area(ABC) 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 Collections: Mathematics