2002 USA TST Problems

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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


Problem 5


Problem 6



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