For each integer $a_0 \gt 1$, define the sequence $a_0$, $a_1$, $a_2$, …, by $$
a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if $\sqrt{a_n}$ is an integer,} \\
a_n + 3 & \text{otherwise}
\end{cases}
$$ for each $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.
A hunter and an invisible rabbit play a game in the plane. The rabbit and hunter start at points $A_0 = B_0$. In the $n$th round of the game ($n \ge 1$), three things occur in order:
(i) The rabbit moves invisibly from $A_{n-1}$ to a point $A_n$ such that $A_{n-1} A_n = 1$.
(ii) The hunter has a tracking device (e.g. dog) which reports an approximate location $P_n$ of the rabbit, such that $P_n A_n \le 1$.
(iii) The hunter moves visibly from $B_{n-1}$ to a point $B_n$ such that $B_{n-1} B_n = 1$.
Let $N = 10^9$. Can the hunter guarantee that $A_NB_N\le100$?
Let $R$ and $S$ be different points on a circle $\Omega$ such that $\overline{RS}$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of $\overline{RT}$. Point $J$ is chosen on minor arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that line $KT$ is tangent to $\Gamma$.
Fix $N \ge 1$. A collection of $N(N+1)$ soccer players of distinct heights stand in a row. Sir Alex Song wishes to remove $N(N-1)$ players from this row to obtain a new row of $2N$ players in which the following $N$ conditions hold: no one stands between the two tallest players, no one stands between the third and fourth tallest players, …, no one stands between the two shortest players. Prove that this is possible.
An *irreducible lattice point* is an ordered pair of integers $(x,y)$ satisfying $\gcd(x,y) = 1$. Prove that if $S$ is a finite set of irreducible lattice points then there exists a nonconstant *homogeneous* polynomial $f(x,y)$ with integer coefficients such that $f(x,y)=1$ for each $(x,y) \in S$.