US-TST 2002

Let $ABC$ be a triangle, and $A$ , $B$ , $C$ its angles. Prove that $$\sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}.$$

Let $p>5$ be a prime number. For any integer $x$ , define $${f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}$$ Prove that for any pair of positive integers $x$ , $y$ , the numerator of $f_p(x) - f_p(y)$ , when written as a fraction in lowest terms, is divisible by $p^3$ .

Let $n$ 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 $A$ and $B$ in $\mathcal S$ such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line $P_1P_n$ ) and $P_1P_n = kAB$ , where (1) $k = 2.5$ ; (2) $k = 3$ .

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

Consider the family of nonisosceles 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 $$\frac{CE}{EM}, \frac{EM}{MC}, \frac{MC}{CE}$$ is constant.

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