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VTMO 2026 - Grade 7 - Semifinal - SET 3 - Self-study

Question 1

A boat travels downstream from $A$ to $B$ in $4$ hours and upstream from $B$ to $A$ in $8$ hours. How many hours does a patch of water hyacinth take to drift from $A$ to $B$?

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  1. $8$
  2. $12$
  3. $4$
  4. $16$
  5. $32$

Question 2

Among the following numbers, which is irrational? $\sqrt{(-2)^4}$; $\sqrt{3\dfrac1{16}}$; $\dfrac{\sqrt{\pi^2}}{|{-\pi}|}$; $\sqrt{2250}$; $(\sqrt3)^4$.
  1. $\sqrt{(-2)^4}$
  2. $\sqrt{3\dfrac1{16}}$
  3. $\dfrac{\sqrt{\pi^2}}{|{-\pi}|}$
  4. $\sqrt{2250}$
  5. $(\sqrt3)^4$

Question 3

Five teams enter a single round-robin football tournament. In each match the winner receives $3$ points and the loser $0$; in a draw each team receives $1$ point. After some matches, no team has $0$ points and all five scores are different. What is the smallest possible number of matches played?

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Question 4

Find the remainder when $2022\times2023\times2024$ is divided by $12$.
  1. $0$
  2. $2$
  3. $4$
  4. $8$
  5. $10$

Question 5

Triangle $ABC$, shown in the figure, has altitudes $AH$ and $BK$. Given $BC=10$ cm, $AH=16.8$ cm and $BK=8$ cm, find $AB$ in centimetres.

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

Question 6

An engine is tested as follows. On the first road segment, the car travels at speed $v$. The second segment is twice as long as the first and is travelled at speed $2v$. The third is three times as long as the first and is travelled at speed $3v$. The car continues in this way: the $n$th segment is $n$ times as long as the first and is travelled at speed $nv$. Find $n$ so that the average speed over all the segments is $7$ times the average speed on the first segment. The speed is constant on each segment.
  1. $7$
  2. $8$
  3. $9$
  4. $11$
  5. $13$

Question 7

The three interior angles of a triangle measure $(6x+5y)^\circ$, $(3x+39)^\circ$ and $(4y+33)^\circ$. Find $x+y$.

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  1. $11^\circ$
  2. $12^\circ$
  3. $13^\circ$
  4. $14^\circ$
  5. $15^\circ$

Question 8

The odd numbers $1,3,5,\ldots,99$ are written on a board. At each turn, erase any two numbers and write the remainder obtained when their sum is divided by $7$. What number remains after $49$ turns?

Question 9

Let $a,b$ be arbitrary positive rational numbers. Which statement below is false?

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  1. $\dfrac{a}{\sqrt{b^4}}$ is rational.
  2. $\dfrac{\sqrt{a^2}}b$ is rational.
  3. $a^2-\sqrt{b^2}$ is rational.
  4. $\sqrt{(-ab)^4}$ is rational.
  5. $\sqrt{\dfrac{a^2b^4}{a^4b}}$ is rational.

Question 10

How many four-digit numbers with different digits, formed from $1,2,3,4,5$, are divisible by $6$?

Question 11

Given $(a+b):(b+c):(c+a)=2:3:5$ and $a+b+c=15$, calculate $a\times b\times c$.

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  1. $0$
  2. $9$
  3. $54$
  4. $120$
  5. $150$

Question 12

Which of the solids below has $12$ edges?

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  1. A
  2. B
  3. C
  4. D
  5. None of the solids.

Question 13

Let $A=11\times12\times13\times\cdots\times20$ and $B=21\times22\times23\times\cdots\times30$. Find the remainder when $B-A-2022$ is divided by $1001$.

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  1. $1$
  2. $11$
  3. $21$
  4. $981$
  5. $991$

Question 14

In triangle $ABC$, $\angle B=40^\circ$ and $\angle C=80^\circ$. A line parallel to $AB$ meets $CA$ and $CB$ at $M$ and $N$, respectively, such that $AM=MN$. Find $\angle BAN$ in degrees.

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Question 15

For natural numbers $a,b$, define $a\star b=(a+1)(b+1)$. Suppose $a>b$ and $a\star b=13$. What is the value of $a$?

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  1. $0$
  2. $1$
  3. $11$
  4. $12$
  5. $13$

Question 16

A basket contains some apples. An takes out one third of them, then returns $15$ apples to the basket. Binh takes one third of the remaining apples, then returns $10$. The numbers of apples taken by An and Binh are equal. Chau also takes one third of the remaining apples, then returns $x$ apples. What is $x$ so that Chau takes the same number of apples as An and Binh?

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  1. $0$
  2. $5$
  3. $8$
  4. $10$
  5. $2$

Question 17

A sequence begins with $1,2,3$. Its fourth term is the sum of the preceding three terms: $1+2+3=6$. Each term from the fourth onwards is likewise the sum of the preceding three terms. What is the eighth term?

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  1. $11$
  2. $20$
  3. $37$
  4. $68$
  5. $99$

Question 18

A digital clock displays the time of day in HH:MM:SS format from 00:00:00 to 23:59:59. How many times in $24$ hours do all six digits change after one second?

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  1. $1$
  2. $2$
  3. $3$
  4. $12$
  5. $24$

Question 19

Let $a,b,c\in\mathbb Z$ satisfy $\dfrac{4^4\times5^5\times6^6}{8^8\times9^9\times10^{10}}=2^a\times3^b\times5^c$. Find $a-b+c$.

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  1. $-37$
  2. $-27$
  3. $-13$
  4. $-3$
  5. $3$

Question 20

Let $A=\overline{abcdef}$ be a six-digit number with $a<b<c<d<e<f$. Find the sum of the digits of $9A$.