problem: Fix integers $a$ and $b$ greater than 1 . For any positive integer $n$, let $r_{n}$ be the (non-negative) remainder that $b^{n}$ leaves upon division by $a^{n}$. Assume there exists a positive integer $N$ such that $r_{n}<2^{n} / n$ for all integers $n \geq N$. Prove that $a$ divides $b$.

problem_match: Problem 4.


problem: Let $B C$ be a fixed segment in the plane, and let $A$ be a variable point in the plane not on the line $B C$. Distinct points $X$ and $Y$ are chosen on the rays $\overrightarrow{C A}$ and $\overrightarrow{B A}$, respectively, such that $\angle C B X=\angle Y C B=\angle B A C$. Assume that the tangents to the circumcircle of $A B C$ at $B$ and $C$ meet line $X Y$ at $P$ and $Q$, respectively, such that the points $X, P, Y$, and $Q$ are pairwise distinct and lie on the same side of $B C$. Let $\Omega_{1}$ be the circle through $X$ and $P$ centred on $B C$. Similarly, let $\Omega_{2}$ be the circle through $Y$ and $Q$ centred on $B C$. Prove that $\Omega_{1}$ and $\Omega_{2}$ intersect at two fixed points as $A$ varies.

problem_match: Problem 5.


problem:
A polynomial $P$ with integer coefficients is square-free if it is not expressible in the form $P=Q^{2} R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $\mathcal{P}_{n}$ be the set of polynomials of the form

$$
1+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}
$$

with $a_{1}, a_{2}, \ldots, a_{n} \in\{0,1\}$. Prove that there exists an integer $N$ so that, for all integers $n \geq N$, more than $99 \%$ of the polynomials in $\mathcal{P}_{n}$ are square-free.

Each problem is worth 7 marks.
Time allowed: $4 \frac{1}{2}$ hours.

problem_match: Problem 6.