VTMO 2026 - Grade 8 - Semifinal - SET 10 - Self-study

Question 1

The cevians $AD,BE,CF$ of triangle $ABC$ concur at $O$. The marked triangle areas are $[AOE]=54$, $[COE]=18$, $[BOD]=100$. All given areas are in square centimetres. Find $[ABC]$ in square centimetres.

Question 2

Find the three-digit password, each digit from 0 to 9. The clues give exact counts: 923 has one correct digit in its correct position; 958 has one correct digit in the wrong position; 379 has two correct digits, both in wrong positions; 142 has no correct digit; 127 has one correct digit in the wrong position.

Question 3

Evaluate $\dfrac{(2102+2012)^2-(2102-2012)^2}{2102\cdot2012}$.

1. 2
2. 4
3. 2012
4. 2102
5. Another answer

Question 4

There are 10 pens, 12 rulers and 14 notebooks. All are packed into 18 gifts, each containing two objects of different types. How many gifts contain a pen and a ruler?

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

Question 5

Numbers $a_1,\ldots,a_n$ are each $1$ or $-1$. Put $S_n=a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1$. Can $S_n=0$ occur when (a) $n=30$; (b) $n=28$? Justify.

Question 6

Fifty students each join at least one of the art, music and science clubs. Every music-club student also joins art. Eight join all three; nine join only science; twenty join music; eighteen join science. How many join only art?

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

Question 7

Distinct natural numbers $x,y$ have greatest common divisor $6$ and ratio $x:y=6:7$. Find $x+y$.

1. 24
2. 42
3. 48
4. 78
5. 26

Question 8

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 $AD/BC$.

Question 9

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

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

Question 10

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?

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 11

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?

Question 12

The four dates $a,b,c,d$ occupy the adjacent calendar cells shown, with $a,b$ in one week and $c,d$ directly below them. If $a+b+c+d=52$, find $a+b$.

1. $19$
2. $25$
3. $26$
4. $27$
5. $32$

Question 13

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

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 14

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?

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

Question 15

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?

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

Question 16

1. $11$
2. $12$
3. $14$
4. $15$
5. $16$

Question 17

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

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

Question 18

1. $45$
2. $210$
3. $609$
4. $1242$
5. Another answer

Question 19

Which 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. $(4,6,8)$
5. $(3,6,9)$

Question 20

1. $(6,8,8)$
2. $(6,9,8)$
3. $(6,10,8)$
4. $(6,11,8)$
5. $(6,12,8)$