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VTMO 2022 — Grade 5 — vtmo

Question 1

$A = 2022 - \left(135 + 357 + 579 + 791 + 20 \times 2 \times 2\right)$. Which calculation equals $A$?
  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

Three equally spaced even natural numbers have mean $2022$. What is the largest possible term?
  1. $0$
  2. $4044$
  3. $2022$
  4. $6066$
  5. $1011$

Question 3

Two times are symmetric when both hour and minute hands reflect across the vertical axis, as illustrated: $10 : 45$ with $1 : 15$; $12 : 55$ with $11 : 05$. How many such pairs in one day are exactly $5$ hours $30$ minutes apart?

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  1. $0$
  2. $1$
  3. $2$
  4. $3$
  5. $4$

Question 4

An has $12$ more candies than the other three’s mean. Binh has $15$ fewer than An, Chau $3$ fewer. After equal redistribution among four, how does Dung’s count change?
  1. Increase $9$
  2. Increase $6$
  3. Increase $12$
  4. Decrease $6$
  5. Increase $15$

Question 5

Using the pictured balances, which object is heaviest? Choices: triangle, pentagon, circle, diamond, cube.

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Question illustration
  1. Triangle
  2. Pentagon
  3. Circle
  4. Diamond
  5. Cube

Question 6

Four black boxes followed by four white boxes are shown. Each move swaps adjacent boxes. What minimum moves make colors alternate?

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Question illustration
  1. $5$
  2. $6$
  3. $7$
  4. $9$
  5. $10$

Question 7

Six pieces (two large triangles, two medium, one small, and one square)form the pictured square. The marked triangle side is $10 \,\mathrm{cm}$. How much does the sum of piece perimeters exceed the outer perimeter?

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  1. $20 \,\mathrm{cm}$
  2. $30 \,\mathrm{cm}$
  3. $100 \,\mathrm{cm}$
  4. $10 \,\mathrm{cm}$
  5. $50 \,\mathrm{cm}$

Question 8

The pictured $I , T , O , L$ tetrominoes use $1 \,\mathrm{cm}^{2}$ cells. Use equal numbers of all four types to tile a square. What is the smallest square’s perimeter?

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Question illustration
  1. $16 \,\mathrm{cm}$
  2. $32 \,\mathrm{cm}$
  3. $48 \,\mathrm{cm}$
  4. $64 \,\mathrm{cm}$
  5. $100 \,\mathrm{cm}$

Question 9

An leaves for school $06 : 30$, studies $4 h = 45 m$, and returns home $12 : 00$. Speeds are $100 m /$ min outward, $80 m /$ min homeward. Find the distance.
  1. $100 m$
  2. $1500 m$
  3. $500 m$
  4. $2000 m$
  5. $1000 m$

Question 10

Split $\frac{21}{22} = \frac{a}{b} + \frac{c}{d}$ witha $/ b , c / d , b / d$ all reduced and $a , b , c , d$ nonzero. Swapping fractions counts as the same decomposition. How many ways?
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $5$

Question 11

Fill $1 \text{--} 9$ once into the pictured $3 \times 3$ grid. The middle row product is $45$ and middle column product $162$. Find the sum of corner entries $A , B , C , D$.

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Question 12

There are $25$ points in the pictured $5 \times 5$ grid. Count squares with grid-point vertices whose every side passes through exactly two grid points (no additional point along the side).

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Question 13

The solid consists of small cubes as pictured. Paint every exterior face, including the bottom. How many cubes receive paint on exactly three faces?

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Question 14

Two overlapping squares have natural-number side lengths that are not consecutive. Region A exceeds $B$ by $39 \,\mathrm{cm}^{2}$ as pictured. Find the larger square’s area.

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Question 15

A teacher gives An $13$ candies, then Binh one-tenth of the remainder, then Chau $16$, then Dung one-tenth of the remainder. An+Binh receive as many as Chau+Dung. Find the initial amount.

Question 16

Two hundred pupils receive symbols $V , T , M , O$, with counts $173 , 161 , 145 , 121$ respectively, and nobody receives all four. How many have both $V$ and $O$?

Question 17

Roll $a = 1 \text{--} 4$ die; odd outcomes move backward, even forward by the shown number. There are $42$ rolls, total $102$ steps in either direction. Counts of $2$ and $4$ are equal; count $1$ is twice count $3$. Find forward steps.

Question 18

$A = 2022$ pair comprises two four-digit numbers, differing by $2022$, one divisible by $5$ and the other by $6$. Order is ignored. How many pairs?

Question 19

Reversing Uncle Binh’s age digits gives An’s age. After three years, their age ratio An: Binh is $3 : 5$. Find An’s current age.

Question 20

Different letters represent different digits, $TITAN + TALENT = SCHOOL$, and $S = 6$. Find $T + I + T + A + N$.

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