Question bank VTMO 2022

What is the last digit of $(1991+1992+\cdots+2022)-(1+2+\cdots+22)+45$?

Four black and four white boxes are arranged as shown. An swaps two boxes each time. What is the minimum number of swaps to make the colors alternate?

The $15$ th of a month is Sunday. Which weekday cannot be the last day of that month?

Two identical tanks: pouring $15$ liters from I into $II$ makes both half full. Pouring $30$ liters from $II$ into I makes tank I full. How many liters were initially in I?

A frog moves from entrance to exit on the grid. It moves horizontally or vertically, cannot cross striped walls, visits each cell at most once, and may move only left, up or down. What is the maximum number of food pieces it can collect?

Uncle An joins boards $I , II , III$ with a striped $60$-cm connector as shown. Board $III$ is twice as long as $II$, and I is twice as long as $III$. How long is the assembled board in centimeters?

Binh scores $3$ for hitting black, $2$ for hitting white, and $- 1$ for missing. He scores $45$ with equal numbers of hits and misses. Which cannot be his number of misses?

Fill the six cells with $1 \text{--} 6$ once each. Any two cells sharing a vertex cannot contain consecutive integers. What is the largest possible number in the ? cell?

An enters the maze at $A$ and must exit at $B$. Reaching a dead end such as $C$ requires returning to the start and counts as a loss. What is the least number of losses needed to guarantee finding the exit?

An attaches small cubes to a cylindrical box as shown, paints their top faces and prints them onto paper. Which picture is the resulting print?

Ten pupils each write two numbers with product $18$. The pairs $2 \times 9$ and $9 \times 2$ are regarded as the same. If every different factor pair occurs, what is the maximum number sharing one answer?

A rectangular paper is divided into two identical rectangles, two identical squares, and a small gray square of area $4 \,\mathrm{cm}^{2}$ as shown. Its side is half the large square’s side. What is the original area in $\,\mathrm{cm}^{2}$?

The digit sum of $2364$ is $15$. Let A be the smallest number with digit sum $2022$. How many digits does A have?

Class $3$ A stands in a line. Viet has three times as many pupils behind him as in front. Nam has six times as many in front as behind. What is the least class size?

Eighty students take attendance twice. In the first round, they repeat the letters $V , T , M , O$ in that order. In the second round, they count from $1$ to $80$. How many students say $O$ in the first round and a number divisible by $3$ in the second round?

An fills the spiral as shown. What number is in the $100$ th gray circle?

An uses three digits to form every $1 - , 2$- and $3$-digit number with no repeated digits. Their total is $5635$. What is the sum of the chosen digits?

Each turn tosses two coins: two down faces give $4$ steps, two up give $1$, mixed faces give $2$. In $30$ turns An moves $69$ steps, with equal counts of the two matching-face outcomes. How many mixed-face turns occur?

The teacher places three differently colored marbles chosen from blue, red, purple, yellow, pink, brown in three positions. A correct color and position earns $3$, correct color in the wrong position $1$, and absent color $0$. The table gives An $3$, Binh $2$, Chau $2$, Dung $6$. Giang guesses red, purple, yellow in that order. What is his score?

Distinct letters represent distinct digits in $\overline{VTMO}+2022=\overline{TITAN}$. Given $O + N = 8$ and odd $M$, find $T + I + T + A + N$.