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VTMO 2026 - Grade 7 - Semifinal - SET 4 - Self-study

Question 1

Start with the first 2019 positive integers. Repeatedly replace each number by the sum of its digits until every resulting number has one digit. What is the sum of the resulting sequence?

Question 2

Hieu has 7 pens, 8 rulers and 9 notebooks. He packs them into 12 gifts, each containing two items of different kinds. How many gifts contain a pen and a ruler?

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Question 3

A security guard works for four days and then takes one day off. Today is Saturday and he is off. What is the least number of days until he next has a Saturday off?

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  1. 7
  2. 12
  3. 28
  4. 34
  5. 35

Question 4

Minh uses each of the digits 1 through 8 exactly once to form two four-digit positive integers. What is the largest possible sum of the two numbers?

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  1. 16173
  2. 15308
  3. 13086
  4. 6912
  5. 16665

Question 5

Triangle $ABC$ is right-angled at $A$, and $AH$ is its altitude to $BC$. Given $AH=1$ cm and $\angle ABC=60^\circ$, find $BC$ in centimetres.

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  1. $\frac{2}{\sqrt3}$
  2. $2\sqrt2$
  3. $2\sqrt3$
  4. $\frac4{\sqrt3}$
  5. $\frac{2\sqrt2}{\sqrt3}$

Question 6

Consider all triangles whose side lengths are natural numbers belonging to $S=\{3,4,6,8\}$. How many different triangles are there?
  1. 3
  2. 4
  3. 16
  4. 17
  5. 20

Question 7

Points $A,B,C$ lie in that order on a straight line, with $AB=2$ and $BC=1$. Points $D,E$ are on the same side of line $BC$, and triangles $ABD$ and $BCE$ are equilateral. Find the area of quadrilateral $ADEC$.

Question illustration
  1. $\frac{\sqrt3}4$
  2. 7
  3. $\frac{3\sqrt3}4$
  4. $\frac{7\sqrt3}4$
  5. $2\sqrt3$

Question 8

Find $a$, given
$$\frac14\times\frac26\times\frac38\times\frac4{10}\times\frac5{12}\times\cdots\times\frac{14}{30}\times\frac{15}{32}=\frac1{2^a}.$$
  1. 16
  2. 17
  3. 18
  4. 19
  5. 20

Question 9

The bar chart shows the number of notebooks sold by a company over several years. In each of 2018 and 2019, sales increased by 60% over the preceding year. In each of 2020 and 2021, sales decreased by 25% from the preceding year. The company sold 30 800 more notebooks in 2021 than in 2017. How many notebooks did it sell in 2021?

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  1. 44000
  2. 70000
  3. 74800
  4. 108000
  5. 119000

Question 10

A box contains 400 marbles coloured blue, white, red, yellow and orange. The ratio of blue to white to red marbles is $1:2:4$. The ratio of white to yellow to orange marbles is $1:3:6$. What is the least number of marbles that must be drawn without looking to guarantee 50 marbles of one colour?

Question 11

For $x\in\mathbb Z$, find the least possible value of
$$|x^2-1024|+|x^2-2024|+|x^2-3024|.$$
  1. 72
  2. 1001
  3. 2001
  4. 2024
  5. 6072

Question 12

In the figure, $BD=10$ cm and $FC=15$ cm. Find the area of rectangle $ADEF$, in square centimetres.

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Question 13

How many minutes after 10:00 will the hour hand and the minute hand first be perpendicular (form a $90^\circ$ angle)?

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  1. 5
  2. $5\frac12$
  3. $5\frac6{11}$
  4. $5\frac5{11}$
  5. Another answer

Question 14

Which of the following numbers has a non-terminating repeating decimal expansion?

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  1. $\frac1{16}$
  2. $\frac1{15}$
  3. $\frac38$
  4. $\sqrt8$
  5. $\frac7{16}$

Question 15

Rectangle $ABCD$ has perimeter 240 cm and $AB=40$ cm. Find the area of the shaded region shown.

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  1. $1000\text{ cm}^2$
  2. $1600\text{ cm}^2$
  3. $2000\text{ cm}^2$
  4. $3200\text{ cm}^2$
  5. $4000\text{ cm}^2$

Question 16

Let $D=\{1,2,3,4,5,6,7\}$. Each operation acts on two numbers: subtract 1 from the first and add 5 to the second. After some operations, the resulting set is
$$D=\{2012,2013,2014,2015,2016,2017,x\}.$$
Which of the following can be the value of $x$?
  1. 2018
  2. 2017
  3. 2019
  4. 2020
  5. 2021

Question 17

Different shapes represent different digits in the addition shown. What digit does the square represent?

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  1. 5
  2. 6
  3. 7
  4. 8
  5. 9

Question 18

In a game, each of four friends $A,B,C,D$ wears a shirt numbered 6, 7 or 8. Anyone wearing 7 must always lie, while anyone wearing 6 or 8 must always tell the truth. Asked for the sum of the four shirt numbers, they reply:

A: “The sum is 28.”

B: “The sum is 27.”

C: “The sum is 26.”

D: “The sum is 25.”
What number is on B's shirt?

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Question 19

Minh's handspan is 2 dm. He measures the floor of a rectangular room and finds its length and width to be 22 handspans and 18 handspans respectively. Find the floor area.
  1. $160\text{ dm}^2$
  2. $200\text{ dm}^2$
  3. $800\text{ dm}^2$
  4. $792\text{ dm}^2$
  5. $1584\text{ dm}^2$

Question 20

The operation $\triangle$ is defined by $a\triangle b=5(a+b)-ab$. For example, $8\triangle5=5(8+5)-8\times5=25$. Given $3\triangle p=31$, find $p$.

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  1. $p=6$
  2. $p=7$
  3. $p=8$
  4. $p=9$
  5. $p=10$