VTMO 2017 — Grade 6 — semifinal

Thanh writes every two-digit number containing digit $6$. How many are there?

$P$ is the product of the first $2017$ primes. Find its last digit.

How many three-digit numbers have distinct digits, digit product $24$ and digit sum $11$?

Three pegs hold A on peg $1 , C$ above $B$ on peg $2$, and $D$ on peg $3$ as shown. Binh removes one disk at a time. How many removal orders are possible?

Distinct natural numbers satisfy $x : y = 6 : 7$ and $\operatorname{gcd} \left(x , y\right) = 6$. Find $x + y$.

$P$ is the smallest composite not divisible by any integer $2 \text{--} 50$. Find its digit sum.

Sixteen lights are off. Each operation toggles exactly three distinct lights. What minimum number of operations turns all on?

The teacher writes $6 , 80 , N$ with $N$ even. Tom reads $6$ as $9$, but obtains the same lcm. Find the least $N$.

A contains natural numbers with distinct digits and digit sum $18$. $D$ is its largest minus its smallest element. Find $D$’s digit sum.

Sixty-four people form a line. Move the first to the back, seat the next, and repeat until one remains standing. Find the survivor’s original position.

Guess a four-digit number: $2896$ has exactly one correct digit, correctly placed; $5273$ has exactly two correct digits, wrongly placed; $1702$ has all four correct but all wrongly placed. Find the number.

An and Binh start together in opposite directions around a park. An takes $12$ minutes per lap, and they first meet after $8$. Find Binh’s lap time.

Two guards begin work May $2 , 2017$. Guard $1$ works six days then rests one; guard $2$ works eleven then rests two, repeating even through holidays. How many days do both rest in $2017$?

Ti can climb $1$ or $2$ steps at a time. How many ways climb $8$ steps?

The pictured tower uses $84$ unit cubes of side $1 \,\mathrm{cm}$. Paint every exterior face except the bottom. Find painted area.