2002 USA TST Problems

Problems from the 2002 USA TST.

Day 1

Problem 1

Let $\displaystyle ABC$ be a triangle. Prove that

$\displaystyle \sin\frac{3A}{2} + \sin\frac{3B}{2} + \sin\frac{3C}{2} \le \cos\frac{A-B}{2} + \cos\frac{B-C}{2} + \cos\frac{C-A}{2}.$


Problem 2

Let $\displaystyle p$ be a prime number greater than 5. For any integer $\displaystyle x$, define

$\displaystyle f_p(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}$.

Prove that for all positive integers $x$ and $y$ the numerator of $\displaystyle f_p(x)-f_p(y)$, when written in lowest terms, is divisible by $\displaystyle p^3$.


Problem 3

Let $\displaystyle n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$ distinct points in the plane. Let $\mathcal S$ denote the union of all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if it is always possible to find points $\displaystyle A$ and $\displaystyle B$ in $\mathcal S$ such that $P_1P_n \mid\mid AB$ (segment $\displaystyle AB$ can lie on line $\displaystyle P_1P_n$) and $\displaystyle P_1P_n = kAB$, where (1) $\displaystyle k = 2.5$; (2) $\displaystyle k = 3$.


Day 2

Problem 4

Let $\displaystyle n$ be a positive integer and let $\displaystyle S$ be a set of $\displaystyle 2^n+1$ elements. Let $\displaystyle f$ be a function from the set of two-element subsets of $\displaystyle S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $\displaystyle x, y, z$ of $\displaystyle S$, one of $\displaystyle f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $\displaystyle a, b, c$ in $\displaystyle S$ such that $\displaystyle f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.


Problem 5

Consider the family of nonisoceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$. Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$. Prove that exactly one of the ratios

$\frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE}$

is constant (i.e., it is the same for all triangles in the family).


Problem 6

Find in explicit form all ordered pairs of positive integers $\displaystyle (m, n)$ such that $\displaystyle mn-1$ divides $\displaystyle m^2 + n^2$.