VTMO 2026 - Grade 8 - Semifinal - SET 08 - Self-study

Question 1

Chau lies on Friday, Saturday and Sunday and tells the truth on the other days. Minh tells the truth on Tuesday, Wednesday and Thursday and lies on the other days. On which day can both say, “Tomorrow I will lie”?

1. Monday
2. Tuesday
3. Thursday
4. Friday
5. Sunday

Question 2

Which segment is the image of $PQ$ under central reflection about $O$, in the given grid?

1. FG
2. TU
3. WV
4. PS
5. MN

Question 3

Four friends A, B, C and D have 76 candies altogether. Simultaneously, A keeps one candy and shares the rest equally with the other three; B similarly keeps two, C three, and D four. Afterwards all four have equal numbers. How many candies did A initially have?

Question 4

Find the units digit of $1\cdot3\cdot5\cdot7+3\cdot5\cdot7\cdot9+5\cdot7\cdot9\cdot11+\cdots+2011\cdot2013\cdot2015\cdot2017$.

1. 0
2. 2
3. 4
4. 6
5. 8

Question 5

A monkey is satisfied if it eats three different kinds of fruit. With 20 apples, 30 pears, 40 bananas and 50 peaches, what is the largest number of monkeys that can be satisfied? Fruits must remain whole.

1. 46
2. 45
3. 44
4. 43
5. 42

Question 6

Quadrilateral $ABCD$ has $AB=BD=BC$ and $\angle DBC=76^\circ$. Find $\angle DAC$ in the shown configuration.

1. 36°
2. 38°
3. 40°
4. 52°
5. 32°

Question 7

Seven unit squares are arranged in two rows inside triangle ABC as shown, four in the lower row and three in the upper row, with the indicated contacts on the triangle’s sides. Find the area of triangle ABC in cm².

1. 16
2. 12.5
3. 10
4. 10.5
5. 15

Question 8

Real numbers $x,y$ satisfy $5x^2+y^2/4=9$. Find the minimum of $S=2xy+1/x^2$ wherever this expression is defined.

1. $\frac59$
2. -7
3. $\frac{-7\sqrt7+4}{7}$
4. 5
5. -5

Question 9

A, B and C are the homes of An, Binh and Chau, collinear in that order; T is their school. AT=6 km, AB=4 km and BC=5 km. Binh leaves at 6:00 and reaches school at 7:00. If Chau also leaves at 6:00 and all walk at the same speed, when does Chau arrive?

1. 7:40
2. 6:40
3. 7:20
4. 7:15
5. 7:30

Question 10

Triangle $ABC$ has area $100$. Points $M,N,P$ lie on $BC,CA,AB$, respectively, with $BM\lt CM$. Triangles $APN,BPM,CMN$ each have area $21$. Find the area of $ABM$.

1. 70
2. 20
3. 21
4. 30
5. 35

Question 11

Parallelogram $ABCD$ has area $24$, and its diagonals meet at $O$. Construct parallelogram $BOCE$. Line $AE$ meets $BC$ at $M$. Find the area of triangle $MCE$.

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

Question 12

The midline of a trapezoid divides it into two parts whose areas are in ratio 2:1. Find the ratio of its larger base to its smaller base.

1. 5:1
2. 4:1
3. 3:1
4. 5:2
5. 3:2

Question 13

On the first day Minh wins 60% of his games against a computer. On the second day he plays ten games and wins all ten, bringing his combined win rate to 70%. What is the minimum number of games he must play on the third day so that his overall win rate is exactly 60%?

Question 14

Evaluate $\dfrac{2024^3-1}{2024^2+2025}$.

1. 2024
2. 2023
3. 1012
4. 1
5. 0

Question 15

A three-digit positive integer has the property that deleting its first digit produces a perfect square and deleting its last digit also produces a perfect square. Find the sum of all such integers.

1. 1013
2. 1177
3. 1465
4. 1993
5. 2016

Question 16

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

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

Question 17

Triangle $ABC$ is right at $A$. Rectangle $ADEF$ has $D$ on $AB$, $F$ on $AC$, and $E$ on $BC$, as shown. Given $BD=10$ cm and $FC=15$ cm, find the rectangle’s area.

(cm²)

Question 18

Given $x+1/y=1$ and $y+1/z=1$, find $xyz$.

1. 1
2. -1
3. -2
4. 2
5. 0

Question 19

(a) Solve over the real numbers: $(x^2-2x+2)^2+(x-1)(3x^2-10x+10)=0$.
(b) Positive integers $x,y,z$, with $x\ne y$, satisfy $x^2+y^2+z^2=189$ and $\dfrac{x}{y}=\dfrac{x^2+z^2}{y^2+z^2}$. Find $xyz$.

Question 20

For parameters $a,b$, the identity $\dfrac{2023x^2+2024x+2022}{x^3-1}=\dfrac{a}{x^2+x+1}+\dfrac{b}{x-1}$ holds wherever defined. Find $ab$.

1. 2022
2. 2023
3. 2024
4. 4048
5. 4088