VTMO 2019 — Grade 7 — final

Six distinct positive integers satisfy $a + b + c = d + e + f = x$. Minimize $x$.

$a$)Given $\frac{bz - cy}{a} = \frac{cx - az}{b} = \frac{ay - bx}{c}$ for nonzero real $a , b , c$, prove $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$. $b$)If six-digit abcdef is divisible by $7$, prove efabcd is divisible by $7$.

Tom trails Jerry by $100 m$ and expects to catch him in $20 s$. At $a = 15 m$ gap, Jerry triples his speed. Tom keeps his speed, and catches him two seconds late. Find Tom’s speed.

$ABC$ has $B = 40 ^{\circ} , C = 20 ^{\circ}$ and angle-A bisector $AD$. Prove $AD = BC - AC$.

$a$)Equal segments $AB , CD$ intersect at $K$ with $AC$ parallel to $BD$. Prove $KA = KC$. $b ) ABC$ is isosceles at $A , BD$ bisects $B , D$ on $AC$, and altitude $AH = \frac{BD}{2}$. Find $A$.

Repeatedly replace each integer $1 , \ldots , 2019$ by its digit sum until one digit remains. Find the final total.