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VTMO 2022 — Grade 4 — vtmo

Question 1

Let $A = 2022 - \left(135 + 357 + 579 + 791 + 20 \times 2 \times 2\right)$. Which calculation has result $A$?
  1. $20 + 2 + 2$
  2. $20 + 2 \times 2$
  3. $20 \times 2 \times 2$
  4. $2 \times 0 + 22$
  5. $20 \times 22$

Question 2

There are $25$ points in a $5 \times 5$ grid. How many squares have vertices among them and at least $3$ grid points on every side?

Question illustration
  1. $15$
  2. $14$
  3. $17$
  4. $16$
  5. $18$

Question 3

Choose pieces to complete the illustrated “VTMO $2022$” picture.

Question illustration

Question illustration
  1. $ADEH$
  2. $CFBG$
  3. $ADBG$
  4. $AFBH$
  5. $CDEG$

Question 4

Two times are symmetric if both hands reflect across the clock’s vertical axis,e.g.$1 : 15$ and $10 : 45$. The pictured clock shows $11 : 10$. What is the least number of minutes until its symmetric time?

Question illustration

Question illustration
  1. $100$
  2. $105$
  3. $110$
  4. $115$
  5. $120$

Question 5

Four black boxes followed by four white boxes are arranged as shown. Each move swaps adjacent boxes. What is the minimum number of moves to alternate colors?

Question illustration
  1. $5$
  2. $4$
  3. $8$
  4. $7$
  5. $6$

Question 6

January $1$ is Monday. Sums of days in any two consecutive months take exactly three consecutive integer values. What weekday is December $31$?
  1. Monday
  2. Tuesday
  3. Wednesday
  4. Thursday
  5. Friday

Question 7

Six wooden pieces (two congruent large triangles, two congruent medium triangles, $a$ small triangle and a square) form the pictured square. The marked large-triangle side is $10 \,\mathrm{cm}$. Find the sum of the six perimeters minus the outer square’s perimeter.

Question illustration
  1. $70 \,\mathrm{cm}$
  2. $90 \,\mathrm{cm}$
  3. $80 \,\mathrm{cm}$
  4. $100 \,\mathrm{cm}$
  5. $60 \,\mathrm{cm}$

Question 8

Two congruent squares and two congruent rectangles fit in a $54 \,\mathrm{cm}$ by $22 \,\mathrm{cm}$ rectangle, leaving the gray square shown. Find its area. The diagram is not to scale.

Question illustration
  1. $225 \,\mathrm{cm}^{2}$
  2. $256 \,\mathrm{cm}^{2}$
  3. $196 \,\mathrm{cm}^{2}$
  4. $289 \,\mathrm{cm}^{2}$
  5. $324 \,\mathrm{cm}^{2}$

Question 9

Among $200$ students, the counts receiving symbols $V , T , M , O$ are $173 , 161 , 145 , 121$; none receives all four. How many have both $V$ and $O$?
  1. $79$
  2. $106$
  3. $133$
  4. $94$
  5. $121$

Question 10

Write $\frac{21}{22} = \frac{a}{b} + \frac{c}{d}$ with nonzero natural $a , b , c , d$; $\frac{a}{b} , \frac{c}{d} , \frac{b}{d}$ all reduced. How many decompositions are there, ignoring order? For example $\frac{1}{2} + \frac{5}{11}$ and $\frac{5}{11} + \frac{1}{2}$ count as one.
  1. $2$
  2. $5$
  3. $1$
  4. $3$
  5. $4$

Question 11

Use the four illustrated piece types to form VTMO. The overlapping parts total $66 \,\mathrm{cm}^{2}$. Find the area of VTMO.

Question illustration

Question 12

Fill the six staircase cells using $1 , 2 , 3 , 4 , 5 , 6$ once each. Cells sharing a vertex cannot contain consecutive numbers. What is the largest possible value at ??

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Question 13

An chooses three distinct digits and forms every $1 - , 2 - , 3$-digit number without repetition within a number. Their sum is $5635$. What is the sum of the chosen digits?

Question 14

$A , B , C , D , E , F , G , H , I , J$ represent distinct digits, with $C , E$ nonzero. Given $\overline{3AB}\times(\overline{CD}\div\overline{EF})\times(G+H-I-J)=2022$, find the maximum $G + H + I + J$.

Question illustration

Question 15

Initially An has $\frac{3}{4}$ as many cards as Binh. An gives Chau some cards, and Binh gives three times that amount, leaving An and Binh equal. Chau receives $48$ in total. How many did An initially have?

Question 16

Hanh rolls a tetrahedral die numbered $1 , 2 , 3 , 4$. Even results move forward by their value; odd results backward by their value. After $42$ rolls, total forward plus backward steps is $102$. Counts of $2$ and $4$ match; count of $1$ is twice that of $3$. How many forward steps are there?

Question 17

After An loses $10$ cards, An has half as many as Binh. Binh then gives An $10$, and Binh’s remaining count is $\frac{7}{5}$ of An’s. What was their initial total?

Question 18

Choose two distinct natural numbers below $222$ whose product is divisible by $77$, but neither number is divisible by $77$. How many unordered pairs are there?

Question 19

A teacher puts three distinct colors from blue, red, purple, yellow, pink, brown in three positions. Correct color+position scores $3$; correct color wrong position $1$; absent color $0$. Guesses: An (blue, red, yellow) scores $3$; Binh (purple, brown, pink)$2$; Chau (blue, yellow, purple)$2$; Dung (pink, brown, yellow)$6$; Giang (red, purple, yellow)?. Find Giang’s score.

Question illustration

Question illustration

Question 20

Different letters represent distinct digits. $TITAN + TALENT = SCHOOL = 639007$. Find $T + I + T + A + N$.