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VTMO 2026 - Grade 8 - Semifinal - SET 04 - Self-study

Question 1

Trapezoid $ABCD$ has $AB\parallel CD$ and $AB/CD=1/4$. Lines $DA$ and $BC$ meet at $I$. Points $M\in AB$, $N\in CD$ satisfy $AM/AB=CN/CD=1/3$. Line $MN$ meets $AD$ and $BC$ at $P,Q$, respectively. Given $IP=IQ$, find $AD/BC$.

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Question 2

Acute triangle $ABC$ has $\angle A=30^\circ$. Altitudes $BE,CF$ meet at $H$. Given $EF=6$ cm, find $BC$ in cm.

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  1. $3\sqrt3$
  2. $4\sqrt3$
  3. $8$
  4. $5\sqrt2$
  5. $6\sqrt2$

Question 3

The diagram is the floor plan of a room. Adjacent walls are perpendicular. The marked lengths are $a,b$. What is the area of the room?

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  1. $2ab+a(b-a)$
  2. $3a(a+b)-a^2$
  3. $3a(b-a)+a^2$
  4. $3ab$
  5. $2(a^2+ab)$

Question 4

Twenty girls stand in a line and four boys join the line. The boys count the girls before them and obtain $17,14,5,2$, respectively. Each girl counts the boys before her. What is the sum of all the girls\' counts?

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Question 5

Simplify $\left(-\frac12a^2b^4c\right)^3:\left(\frac14ab^2c\right)^2$.

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  1. $2a^4b^2c^8$
  2. $-2a^4b^6c^2$
  3. $-2a^4b^2c^8$
  4. $-2a^4b^8c$
  5. $-2a^4b^8c^2$

Question 6

Hai scored $50\%$ of the total marks on a test. He answered $13$ of the first $20$ questions correctly. Running out of time, he guessed the rest and answered $25\%$ of those correctly. How many questions were on the test?

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  1. 36
  2. 38
  3. 24
  4. 32
  5. 40

Question 7

In parallelogram $ABCD$, diagonals meet at $O$. Let $E$ lie on $OD$, and let $F$ be the reflection of $C$ in $E$. Where must $E$ lie for $AFDE$ to be a parallelogram?

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  1. $DE=DO$
  2. $DE=\frac23DO$
  3. $DE=\frac12OE$
  4. $DE=\frac12DO$
  5. $DE=\frac13DO$

Question 8

Real numbers $a,b,c$ satisfy $a+b+c=10$ and $ab+bc+ca=15$. Find $a^2+b^2+c^2$.

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  1. 70
  2. 65
  3. 60
  4. 75
  5. 85

Question 9

Which triple can be the lengths of the three altitudes of a nondegenerate triangle?

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  1. $(1,2,3)$
  2. $(1,0.5,1.5)$
  3. $(2,4,4)$
  4. $(4,6,8)$
  5. $(3,6,9)$

Question 10

Square $ABCD$ has side $9$ cm. Points $M,N,P,Q$ lie on $AB,BC,CD,DA$, with $AM=BN=CP=DQ=3$ cm. Let $E,H$ be the intersections of $AC$ with $MQ,NP$, respectively, and $F,K$ the intersections of $BD$ with $MN,PQ$. Find the area of square $EFHK$.

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Question 11

Triangle $ABC$ is isosceles at $A$. Point $O$ is inside it, with $OA=3$ cm and $OB=OC=2$ cm. The distance from $O$ to $AB$ is $1$ cm. Find the distance from $O$ to $BC$.

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  1. $\frac{2\sqrt6+1}{3}$
  2. $\frac{\sqrt6-1}{3}$
  3. $\frac{2\sqrt6-1}{3}$
  4. $\frac{\sqrt6-1}{2}$
  5. $\frac{\sqrt6+1}{2}$

Question 12

The first square has area $64$ cm². The diagonal of the second equals the side of the first, the diagonal of the third equals the side of the second, and so on. What is the area of the ninth square?

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  1. $\frac18$
  2. $\frac14$
  3. $\frac13$
  4. $\frac12$
  5. $\frac19$

Question 13

How many integers $n$ make $2^{2020}/(3n-1)$ an integer?

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  1. 2018
  2. 2019
  3. 2020
  4. 2021
  5. 2022

Question 14

A set $S$ contains distinct positive integers, no two of which are twice one another. Its smallest element is $5$ and its largest is $1023$. What is the greatest possible number of elements?

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  1. 680
  2. 768
  3. 512
  4. 678
  5. 350

Question 15

A corridor has $46$ lights, initially all off. Each move changes the state of any three lights: off becomes on and on becomes off. What is the least number of moves to turn all $46$ lights on?

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  1. 15
  2. 16
  3. 17
  4. 18
  5. 19

Question 16

Given $x+y=5$ and $x^2+y^2=111$, find $x^3+y^3$.

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  1. 115
  2. 227
  3. 300
  4. 555
  5. 770

Question 17

Right triangle $ABC$ has its right angle at $A$, with $AB=7$ cm and $AC=14$ cm. A square $MNPQ$ is inscribed with $M\in AB$, $N\in AC$, and $P,Q\in BC$, as shown. Find its area in cm².

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Question 18

Hai\'s salary is $5/8$ of Tuan\'s. Hai saves $40\%$ of her salary, and her spending is half of Tuan\'s spending. What percentage of his salary does Tuan save?

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  1. 20%
  2. 25%
  3. 30%
  4. 35%
  5. 40%

Question 19

An and Binh start simultaneously from two places, moving toward one another at constant planned speeds, and would meet after $5$ hours. After $1$ hour An stops for $30$ minutes, then resumes at $80\%$ of his original speed. Binh keeps his speed. They actually meet $6$ hours after starting. Find the ratio of An\'s original speed to Binh\'s.

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  1. $\frac25$
  2. $\frac52$
  3. $\frac13$
  4. $3$
  5. $\frac53$

Question 20

Start with the set $\{1,2,3,4,5,6,7\}$. At each step choose two numbers, subtract $1$ from one and add $5$ to the other. After some steps the resulting set is $\{2012,2013,2014,2015,2016,2017,x\}$. Which listed value is $x$?

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  1. 2018
  2. 2017
  3. 2019
  4. 2020
  5. 2021