VTMO 2026 - Grade 8 - Semifinal - SET 09 - Self-study

Question 1

A teacher draws a quadrilateral. An calls it an isosceles trapezoid; Binh a parallelogram; Cuong a rectangle; Danh a square; Minh a right trapezoid. Exactly one student is incorrect. Who is it?

1. An
2. Binh
3. Cuong
4. Danh
5. Minh

Question 2

For every natural number $a$, prove that $A=(a+2013)(a+2011)(a+2009)(a+2007)+16$ is a perfect square.

Question 3

Hanh and Phuc must fold 16 paper planes and 10 paper cranes. Hanh needs 2 minutes per plane and 6 per crane; Phuc needs 3 per plane and 5 per crane. Working together, what is the shortest time to finish all the figures?

1. 43
2. 45
3. 50
4. 40
5. 44

Question 4

At 6:00 two cars start from A and B towards each other at constant speeds. They meet at 7:12 and continue. The first car reaches B at 8:00. When does the second reach A?

1. 8:00
2. 9:00
3. 10:00
4. 11:00
5. 8:30

Question 5

Triangle $ABC$ is right at $A$, and $AH$ is the altitude to $BC$. Given $HB=3$ cm and $HC=12$ cm, find its area in cm².

1. 36
2. 40
3. 45
4. 50
5. 60

Question 6

An even positive integer $n$ has exactly $15$ positive divisors. What is the greatest possible number of positive divisors of $3n/2$?

Question 7

A 5×5 board has one light in each cell. Pressing a cell toggles its light and the lights in cells sharing a side with it. Initially black cells are off and white cells are on, as shown. Can all 25 lights be turned off? Justify.

Question 8

Square $ABCD$ has side $3$. Points $M,N,P,Q$ lie on $AB,BC,CD,DA$ with $AM=BN=CP=DQ=1$. Diagonal $BD$ meets $MN$ at $E$ and $PQ$ at $F$. Find $EF$.

1. $\frac{3\sqrt3}{2}$
2. $\sqrt3$
3. $\sqrt2$
4. $\frac{4\sqrt2}{3}$
5. $\frac{5\sqrt2}{3}$

Question 9

The odd positive integers are grouped as {1}, {3,5}, {7,9,11}, {13,15,17,19}, and so on. What is the middle number in the fifteenth group?

1. 113
2. 209
3. 211
4. 215
5. 225

Question 10

Triangle $ABC$ has $AB=5$, $AC=7$, $BC=8$. The internal angle bisectors $BE,CF$ meet at $I$. Points $M,N$ lie on $BC$ with $BM=5$, $CN=7$. Lines $ME,NF$ meet at $K$, and $IK$ meets line $AB$ at $D$.
(a) Prove $\angle ABC=60^\circ$.
(b) Find $AD$.

Question 11

Find the units digit of $1!+2!+3!+\cdots+2014!$.

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

Question 12

Let $P(x)=x^2/(x^2-10x+50)$. Find $P(1)+P(2)+\cdots+P(10)$.

1. 5
2. 9
3. 10
4. 11
5. 12

Question 13

Rectangle $ABCD$ has area $120$. Its diagonals meet at $O$. Point $E$ is the midpoint of $OC$, and $DE$ meets $BC$ at $I$. Find the area of quadrilateral $OBIE$.

1. 25
2. 30
3. 35
4. 36
5. 40

Question 14

Equilateral triangle $AMN$ is inscribed in unit square $ABCD$, with $M$ on $BC$ and $N$ on $CD$ as shown. Find its area.

1. $-3+2\sqrt3$
2. $4-2\sqrt3$
3. $\frac12$
4. $\frac{\sqrt3}{3}$
5. $\frac{\sqrt3}{4}$

Question 15

Choose 100 distinct positive integers. A pair is called nice if the larger number is twice or three times the smaller. What is the largest possible number of nice pairs? A number may belong to several pairs.

Question 16

For natural $n$, consider
$A=37^n+71^n-2^n-1$;
$B=n(n^2-6)^2(n^2-1)-n^3(n-1)(n+1)$;
$C=1+2+\cdots+(21n+1)$.
Which expressions are always divisible by $21$?

1. Only A
2. A and B
3. B and C
4. Only C
5. All three

Question 17

A right triangle has shortest side 3 cm and smallest angle 15°. Find the length of the other leg.

1. $3(2+\sqrt3)$
2. $6+\sqrt3$
3. $3+3\sqrt3$
4. $2+3\sqrt3$
5. $2+\sqrt3$

Question 18

Evaluate $\dfrac{3^3+1}{2^3-1}\cdot\dfrac{4^3+1}{3^3-1}\cdots\dfrac{10^3+1}{9^3-1}$.

1. 150
2. 155
3. 160
4. 165
5. 170

Question 19

Write the integers from 1 through 2014 consecutively with no separators. How many digits are written?

1. 6760
2. 6234
3. 6949
4. 6849
5. 6939

Question 20

Find the greatest natural number $k$ such that $k^2$ divides $10!$.

1. 256
2. 360
3. 512
4. 625
5. 720