VTMO 2016 — Grade 7 — final

$A = 3 \times 4$ grid has $- 8 , 1 , - 1 , - 2$ prefilled as shown. Remove one of $0 , 2 , 3 , 4 , 5 , 7 , 8 , 10 , 11$, and fill the eight remaining cells so every row and column has the same average. Explain.

$a ) 25 \%$ of entrants win, $\frac{2}{7}$ of boys win, $420$ boys do not, and $648$ girls participate. How many girls win? $b$)Find the least positive multiple of $999$ containing no digit $9$.

Two riders start simultaneously from $A , B$ toward each other, turn immediately at the other endpoint, and meet first $15 \,\mathrm{km}$ from $A$, then $9 \,\mathrm{km}$ from $B$ while traveling opposite ways. Find $AB$.

$A , B , C , D$ wear $6 , 7$, or $8$. $A = 7$ always lies, $a = 6$ or $8$ tells truth. Asked their total, A says $28 , B = 27 , C = 26 , D = 25$. Find each shirt number.

$M$ is the midpoint of $BC , N$ the midpoint of $AM$. $D$ is opposite $B$ across $N$, with $DN = NB$. Let $E$ be the intersection of rays $CD$ and $BA$. Prove $CD = AM$ and $AB = AE$.

$AB \lt AC$. $E$ lies beyond $B$ on ray $AB$ with $BE = AC$; $F$ beyond $C$ on ray $AC$ with $CF = AB$. $M$ is midpoint $BC , N = AM \cap EF , D = NC \cap EM$. Prove $MA = MN$ and $AN = DF$.