VTMO 2017 — Grade 6 — final

Fill $1 \text{--} 9$ once each in $a = 3 \times 3$ grid so every row, column, and both diagonals sum to a multiple of $9$. $a$)Find all center values $T$. $b$)Give a grid for each.

$a ) 14 , 53 , 79$ give equal remainders modulo $N$. Find the maximum $N$. $b$)A four-digit number abab has exactly six positive divisors. Find the smallest.

Initially Titan grade $6$ has boys: girls $4 : 3$. Added girls change the ratio to $3 : 5$; added boys then give $8 : 7$. Two fewer boys than girls have joined altogether, and nobody leaves. Find the final enrollment.

An travels $600 m$, spending half the time running $10 \frac{\,\mathrm{km}}{h}$ and half walking $5$. Binh runs half the distance and walks half at $5$, finishing in the same time. Find Binh’s running speed.

Increasing a cube’s height by $3 \,\mathrm{cm}$ produces a cuboid whose surface area increases $72 \,\mathrm{cm}^{2}$. Find the original volume.

Two football teams have ten players each, one all truthful, one all liars. Asked their goals, the answers are $1 , 2 , 3 , 5$, with only Chau answering $2$. The truthful team wins $20 : 17$. Is Chau truthful or lying?