VTMO 2015 — Grade 7 — final

Permute $1 , \ldots , 2015$ and subtract $i$ from the entry at position $i$. Let $P$ be the product of these $2015$ differences. Prove $P^{2} + 2 P$ divisible by $8$.

$S \left[A\right]$ sums the element sums of all subsets, including the empty subset. Find $S \left[\{ 1 , 2 , 3 \}\right]$, and prove $S \left[\{ a , b , c , d \}\right]$ divisible by $8$ for distinct natural elements.

$ABC$ is right at A with $AB \lt AC$. $M$ is the midpoint of $BC$; $D$ lies opposite A across $M$. $E , F$ are projections of $B , C$ onto $AD$. Prove $BFD = CEA$. Let $DK$ be perpendicular to $BC$ and $N = DK \cap BE$; prove $AN = NC$ and find $ANC$ when $B = 60 ^{\circ}$.

If $\frac{a}{b} = \frac{c}{d}$, prove $\frac{a^{2} + b^{2}}{c^{2} + d^{2}} = \frac{ab}{cd}$ where defined. Also minimize $\frac{6}{| x | - 3}$ for integer $x$.

Tanker $B$ starts $100 \,\mathrm{km}$ north of cargo ship $A$. A sails northwest at $20 \frac{\,\mathrm{km}}{h}$ and $B$ southeast at $30 \frac{\,\mathrm{km}}{h}$. How far has A traveled when they are closest?