VTMO 2017 — Grade 7 — final

$M$ uses digits $1 \text{--} 8$ once. Every length $- k$ prefix is divisible by $k$, for $k = 2 , \ldots , 8$. Find $M$.

$a ) A = 0.1 ( 2453$), with the parentheses denoting repetition. Find the least positive $k$ making kA integral. $b$)The six pairwise gcds of four distinct positive integers are $1 , 2 , 3 , 4 , 5 , N , N \gt 5$. Minimize $N$.

After $70 \,\mathrm{km}$ at speed $v , a$ train would arrive $1 h = 20$ min late if speed were reduced $25 \%$. Traveling another $40 \,\mathrm{km}$ at $v$ before reducing gives only $1 h$ lateness. Find $v$.

A alone needs $20$ days, $B = 30$. Working together, A rests three days and $B$ two, with no common rest days. How long does completion take?

$ABC$ is isosceles at $A , BD = \frac{BC}{3}$. $E$ is beyond $D$ on $AD$ with $DE = AD$; $F$ is beyond $E$ on $BE$ with $EF = BE$. Prove $FD$ perpendicular to $BC$, and find $FD$ for equilateral $ABC$ with side $1$.

$a$)Two ten-player teams are entirely truthful or entirely lying. One player reports one goal, Chau two, one three, and everyone else five. The truthful team wins $20 : 17$. Is Chau truthful? $b ) A =$ abcdef has increasing digits. Find the digit sum of $9 A$.