USAMO 2006

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and

$$\left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p}$$

if and only if $s$ is not a divisor of $p-1$ .

Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ , and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

For a given positive integer $k$ find, in terms of $k$ , the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$

For integral $m$ , let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence $$\{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0}$$ is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$ )

Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$

A mathematical frog jumps along the number line. The frog starts at $1$ , and jumps according to the following rule: if the frog is at integer $n$ , then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$ , respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$ . Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ , respectively. Prove that the circumcircles of triangles $SAE$ , $SBF$ , $TCF$ , and $TDE$ pass through a common point.