VTMO 2014 — Grade 7 — final

Given $a + b + c = 3 , ab + bc + ca = 1 , abc = - 1$, prove exactly one of $a , b , c$ equals $1$.

Vy travels $V$→$B$ on flat/uphill/downhill sections at $\frac{\frac{5}{4}}{6} \frac{\,\mathrm{km}}{h}$, taking $1 h = 36$ min; the return takes $1 h = 39$ min. The flat section is $2.5$ km. Find the route length.

Prove $S = \frac{1}{2} + \frac{2}{2^{2}} + \ldots + \frac{2014}{2^{2014}} \lt 2$.

$ABC$ is right-angled at $A$. Bx bisects angle $B$; $D$ is the perpendicular projection of $C$ onto Bx. $M$ is the midpoint of $AC$, and $DM$ meets $BC$ at $N$. Prove $DAC$ is isosceles, $N$ is the midpoint of $BC$, and find $B$ when $BD = AC$.

Students stand in a circle. Candy shares increase $1 , 2 , 3 , \ldots$ continuously into the second round. Round $2$ contains $100$ more candies than round $1$. How many students are there?