US-TST 2003

For a pair of integers $a$ and $b$ , with $0 < a < b < 1000$ , set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a skipping set for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$ , $|s_1 - s_2|\not\in \{ a, b\}$ . Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$ . Determine the maximum and minimum values of $f$ .

Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$ , $PB$ , $PC$ intersect sides $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ , respectively. Prove that $$[PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC]$$ if and only if $P$ lies on at least one of the medians of triangle $ABC$ . (Here $[XYZ]$ denotes the area of triangle $XYZ$ .)

Find all ordered triples of primes $(p, q, r)$ such that $$p \mid q^r + 1, q \mid r^p + 1, r \mid p^q + 1.$$ Reid Barton

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$f(m+n)f(m-n) = f(m^2)$$ for $m,n \in \mathbb{N}$ .

Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$ . Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$ .

Let $\overline{AH_1}, \overline{BH_2}$ , and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$ . The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$ , respectively. For $k = 1, 2, 3$ , let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$ ) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$ . Prove that the circumcircles of triangles $T_1P_1T_2$ , $T_2P_2T_3$ , $T_3P_3T_1$ pass through a common point.