Find all positive integers $m$ and $n$ such that $1 + 5 \cdot 2^m = n^2$.

Let $f$ be a function defined on the positive integers, taking positive integral values, such that $f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$, $f(a) < f(b)$ if $a < b$, $f(3) \geq 7$. Find the smallest possible value of $f(3)$.

Let $PRUS$ be a trapezium such that $\angle PSR = 2\angle QSU$ and $\angle SPU = 2 \angle UPR$. Let $Q$ and $T$ be on $PR$ and $SU$ respectively such that $SQ$ and $PU$ bisect $\angle PSR$ and $\angle SPU$ respectively. Let $PT$ meet $SQ$ at $E$. The line through $E$ parallel to $SR$ meets $PU$ in $F$ and the line through $E$ parallel to $PU$ meets $SR$ in $G$. Let $FG$ meet $PR$ and $SU$ in $K$ and $L$ respectively. Prove that $KF$ = $FG$ = $GL$.

There are $n$ points on a circle, such that each line segment connecting two points is either red or blue. $P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$. (a) For which values of $n$ is this possible? (b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.

In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.

For any positive integer $n$, define $a_n$ to be the product of the digits of $n$. (a) Prove that $n \geq a(n)$ for all positive integers $n$. (b) Find all $n$ for which $n^2-17n+56 = a(n)$.