a) In a competition, $25\%$ of all contestants receive an award. Of the male contestants, $\frac27$ receive an award, while $420$ males do not. There are $648$ female contestants. How many females receive an award?
b) Find the smallest positive integer that contains no digit $9$ and is divisible by $999$.
Triangle $ABC$ is right-angled at $A$. The ray $Bx$ is the internal bisector of $\angle A$ [the source angle label is preserved]. From $C$, draw the perpendicular to $Bx$, meeting it at $D$. Let $M$ be the midpoint of $AC$, and let $DM$ meet $BC$ at $N$.
A train travels from $A$ to $B$ at a constant speed. After $70$ km, the driver discovers an engine fault and proposes reducing the speed by $25\%$. The conductor calculates that this would make the train arrive $1$ hour $20$ minutes late, beyond the permitted limit. He therefore asks the driver to continue at the original speed for another $40$ km before making the proposed reduction. The train then arrives $1$ hour late. Find its originally intended speed.
The set $A=\{1,2\}$ has four subsets: $\varnothing$, $\{1\}$, $\{2\}$, $\{1,2\}$. Their element sums are $0,1,2,3$. Write $S[A]$ for the sum of the element sums over all subsets of $A$; thus $S[A]=0+1+2+3=6$.
a) Let $B=\{1,2,3\}$. Find $S[B]$.
b) Let $C=\{a,b,c,d\}$, where $a,b,c,d$ are distinct nonnegative integers. Prove that $S[C]$ is divisible by $8$.