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 $p-1$.
Proposed by Kiran Kedlaya (credited in the supplied source).
Let $k > 0$ be a fixed integer. Compute the minimum integer $N$ (in terms of $k$) for which there exists a set of $2k+1$ distinct positive integers that has sum greater than $N$, but for which every subset of size $k$ has sum at most $N/2$.
Proposed by Dick Gibbs (credited in the supplied source).
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
$$
\{ p(f(n^2)) - 2n \}_{n \geq 0}
$$
is bounded above. (In particular, this requires $f(n^2) \neq 0$ for $n \ge 0$.)
Proposed by Titu Andreescu, Gabriel Dospinescu (credited in the supplied source).
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 $k \ge 2$ 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 $2^i$.
Proposed by Zoran Sunik (credited in the supplied source).
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.
Proposed by Zuming Feng, Zhonghao Ye (credited in the supplied source).