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VTMO 2026 - Grade 7 - Semifinal - SET 1 - Self-study

Question 1

Quadrilateral $ABCD$ has side lengths $AB=11$ cm, $BC=7$ cm, $CD=9$ cm and $DA=3$ cm. The angles at $A$ and $C$ are right angles. Find its area.

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  1. $30\text{ cm}^2$
  2. $44\text{ cm}^2$
  3. $48\text{ cm}^2$
  4. $60\text{ cm}^2$
  5. $96\text{ cm}^2$

Question 2

The two diagonals of a $3\times3$ square grid pass through $5$ small squares, as shown. How many small squares do the two diagonals of an $11\times11$ square grid pass through?

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  1. $19$
  2. $20$
  3. $21$
  4. $22$
  5. $23$

Question 3

Calculate $(1+1)\times\left(1+\frac12\right)\times\left(1+\frac13\right)\times\left(1+\frac14\right)\times\cdots\times\left(1+\frac1{100}\right)$.
  1. $\frac{99}{100}$
  2. $\frac{101}{100}$
  3. $\frac1{100}$
  4. $100$
  5. $101$

Question 4

Bình takes a test with two parts: Part 1 contains $10$ multiple-choice questions and Part 2 contains $30$ fill-in-answer questions. He answers $80\%$ of Part 1 and $70\%$ of Part 2 correctly. What percentage of all the test questions does he answer correctly?

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  1. $72.5\%$
  2. $73\%$
  3. $73.5\%$
  4. $74\%$
  5. $76\%$

Question 5

Which of the following products is a perfect square? The factorial notation is illustrated by $1!=1$, $2!=2\times1=2$, $3!=3\times2\times1=6$, and $4!=4\times3\times2\times1=24$.
  1. $98!\times99!$
  2. $98!\times100!$
  3. $99!\times100!$
  4. $99!\times101!$
  5. $100!\times101!$

Question 6

For a positive integer $n$, let $T=\frac1{2\times5}+\frac1{5\times8}+\cdots+\frac1{(3n-1)(3n+2)}$. Which expression below is the simplified form of $T$?

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  1. $\frac{3n}{2(3n+2)}$
  2. $\frac{n}{3n+2}$
  3. $\frac{n}{2(3n+2)}$
  4. $\frac{3n}{2(2n+3)}$
  5. $\frac{3n}{2(n+3)}$

Question 7

In triangle $ABC$, $AB<AC$, $\angle B=70^\circ$, and the exterior angle at $C$ is $130^\circ$. Point $D$ lies on $AC$ with $AD=AB$. Find $\angle BDC$.

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  1. $50^\circ$
  2. $60^\circ$
  3. $70^\circ$
  4. $120^\circ$
  5. $130^\circ$

Question 8

The product of two positive integers is $10000$. Neither integer is divisible by $10$. Find their positive difference.
  1. $45$
  2. $210$
  3. $609$
  4. $1242$
  5. Another answer

Question 9

In triangle $ABC$, point $E$ lies on $BC$ with $BE=2EC$, and point $M$ lies on $AE$ with $ME=3MA$. Line $BM$ meets $AC$ at $D$. Given that triangle $MEC$ has area $297\text{ cm}^2$, find the area of triangle $MAD$, in $\text{cm}^2$.

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  1. $17$
  2. $18$
  3. $19$
  4. $20$
  5. $21$

Question 10

Calculate $\displaystyle \frac8{1+\frac{20}{23}}\div\frac8{1-\frac{20}{23}}$.
  1. $\frac3{43}$
  2. $\frac{20}{23}$
  3. $\frac{23}{20}$
  4. $1$
  5. $\frac{129}{529}$

Question 11

Find the smallest positive integer $p$ such that $540p$ is a perfect square.
  1. $2$
  2. $3$
  3. $5$
  4. $15$
  5. $60$

Question 12

Given $\frac{a+b}{2c}=\frac{a-b}{2d}=\frac{c+d}{2a}=\frac{c-d}{2b}$, calculate
$$S=\frac{(a+b)^2-(a-b)^2+(c+d)^2-(c-d)^2}{a^2-b^2+c^2-d^2}.$$

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Question 13

How many three-digit numbers are divisible by $3$ and have digits that are either all even or all odd?

Question 14

A palindrome is a natural number that reads the same forwards and backwards. For example, $1221$ is a palindrome. How many three-digit palindromes are there?

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  1. $19$
  2. $90$
  3. $100$
  4. $900$
  5. $450$

Question 15

The number $20142014\ldots2014$ contains $1500$ digits. What is the greatest number of digits that can be deleted so that the remaining number has digit sum $2014$?
  1. $860$
  2. $868$
  3. $996$
  4. $611$
  5. $375$

Question 16

Let $M$ be a point inside equilateral triangle $ABC$. Let $D,E,F$ be the perpendicular projections of $M$ onto $BC,CA,AB$, respectively. Given $MD=1$, $ME=2$ and $MF=3$:

a) Find the area of triangle $ABC$.

b) The line through $B$ perpendicular to $AC$ meets $AC$ at $H$. On $BH$, choose $G$ so that $BG=2GH$. Prove that $GM\parallel AC$.

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Question 17

How many months in $2020$ have exactly four Saturdays? You are given that 1 January 2020 was a Wednesday.

Question 18

In a group of $50$ students, every student belongs to at least one of three clubs: Art, Music and Science.

Every Music Club member also belongs to the Art Club. Eight students belong to all three clubs. Nine belong only to Science. The Music Club has $20$ members and the Science Club has $18$ members. How many students belong only to Art?

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  1. $16$
  2. $18$
  3. $19$
  4. $20$
  5. Another answer

Question 19

Alcohol strength in degrees describes the volume of pure alcohol in $100$ ml of a mixture of pure alcohol and water. For example, Bầu Đá liquor at $55^\circ$ contains $55$ ml of pure alcohol and $45$ ml of water per $100$ ml.

A mixer wants to reduce the strength of a bottle from $55^\circ$ to $T^\circ$. Either adding $v$ units of water or removing $v$ units of pure alcohol from the original bottle gives that same strength $T^\circ$. Find $T$.

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Question 20

Nam wants a bouquet containing only red, white and yellow roses. How many ways can he choose a bouquet of $7$ roses with all three colours present?

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  1. $12$
  2. $13$
  3. $14$
  4. $15$
  5. $16$