VTMO 2022 — Grade 6 — vtmo

Among the fractions $\frac{n}{99 - n} , n = 1 , \ldots , 98$, how many are written with exactly two distinct digits, without reducing them?

Natural numbers $a , b , c$ satisfy $\operatorname{gcd} \left(a , b\right) = 30 , \operatorname{lcm} \left(a , b\right) = 180 , \operatorname{gcd} \left(a , b , c\right) = 15$ and $\operatorname{lcm} \left(a , b , c\right) = 900$. Minimize $a + b + c$.

Find the sum of the last two digits of $987^{2022} - 625^{2022} - 387^{2022}$.

A square inscribed in a circle of diameter $22 \,\mathrm{cm}$ is divided into eight congruent rectangles as shown. Find the perimeter of the bold square joining the side midpoints.

Notebook sales increased $60 \%$ annually in $2018$ and $2019$, then decreased $25 \%$ annually in $2020$ and $2021$. Sales in $2021$ exceeded $2017$ by $30800$. How many notebooks were sold in $2021$?

How many two-digit numbers $ab$ have both $a + b$ and $a - b$ prime?

A square card is divided into five squares, two congruent, and the hatched rectangle shown. The dotted square has area $4 \,\mathrm{cm}^{2}$. Find the hatched rectangle area.

How many subsets of $\{ 1 , 2 , 3 , 4 , 5 , 6 \}$ have at least two elements, all multiples of their minimum?

Father and mother average $32$ years, An and Binh average $8$. Father and Binh have the same average age as mother and An. How much older is mother than Binh?

Fill $0$ through $11$ once into the diagram so all square vertex sums are equal. Given $0 , 1 , 2 , 3$, find“?”.

Chau paints the entire outside, including the bottom, of the unit-cube solid shown. Each small face needs $1$ ml. Find the paint required.

Wheels $A$ and $C$ contain $- 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4$; $B$ chooses+ or−. How many spin-outcome triples yield a positive integer?

An alone takes the same time as Binh and Chau together. Binh alone takes twice the time of Chau and An together. Chau alone takes $t$ times the time of An and Binh together. Find $t$.

Place $1 \text{--} 9$ once into $a = 3 \times 3$ grid. Read three rows, three columns and two diagonals left-to-right/top-to-bottom as eight three-digit numbers. Minimize their sum.

A standard die has opposite faces summing $7$. It rolls along the pictured arrows without lifting or twisting. Find $a + b + c$ on the final visible faces.

The houses of An, Binh, Hanh, Phuc and school occur in the order shown. Phuc lives $600 m$ from school. They start together and each needs $15$ minutes at a constant individual speed. When Phuc has covered one third of the journey, the others reach the next house. How far is An from school?

An moves up/right from A to $B$ and Binh down/left from $B$ to A on the pictured $3 \times 3$ grid. They start together at equal speeds. How many route choices lead to their meeting?

The pictured exact division of $V = 100$ by $VT$ has quotient $MO$, with subtractions $1 IE$ and $1 F = 0$. Different letters represent different digits; $1$ and $0$ in $V = 100$ are fixed numerals. Find $VIET$+VTMO.

Let $A = 1 \times 2 \times \ldots \times 10$ and $B = 11 \times 12 \times \ldots \times 20$. Find the remainder of $B - A - 22$ modulo $21$.

Parallelogram $ABCD$ has $AE = \left(\frac{2}{3}\right) AB , CF = \left(\frac{2}{3}\right) CD$ and area $600 \,\mathrm{cm}^{2}$. $H = AC \cap DE$ and $K = AC \cap DB$ as shown. Find triangle $FHK$ area.