VTMO 2026 - Grade 7 - Semifinal - SET 8 - Self-study

Question 1

Let $\angle xOy=90^\circ$. Inside it, draw rays $Oz,Ot$ such that $\angle xOz=30^\circ$ and $\angle tOy=30^\circ$. Take $K$ on ray $Ox$ with $OK=\sqrt3$. From $K$, draw the perpendicular to $Oz$, meeting it at $B$. Through $B$, draw the perpendicular to $Ot$, meeting ray $Oy$ at $D$. Find $OD$.

1. $\frac12$
2. $\frac2{\sqrt3}$
3. $2\sqrt3$
4. $\frac32$
5. $\frac{\sqrt3}2$

Question 2

The sums of the numbers in all four $2\times2$ squares of the displayed $3\times3$ grid are equal. Find $x$.

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

Question 3

Find the sum of the five marked angles $a,b,c,d,e$ in the star.

1. $150^\circ$
2. $180^\circ$
3. $210^\circ$
4. $240^\circ$
5. $360^\circ$

Question 4

Let $n$ be a natural number with $n<2023$. Find the largest possible $n$ such that the decimal representation of $9999n$ contains no digit 9.

Question 5

Parallelogram $ABCD$ has area $144\text{ cm}^2$. Point $M$ lies on $AB$. Let $O=AC\cap BD$, $E=DM\cap AC$ and $F=CM\cap BD$, as shown. The two hatched triangles $ADE$ and $BCF$ have total area $48\text{ cm}^2$. Find the area of quadrilateral $MFOE$.

1. 36
2. 18
3. 12
4. 9
5. 24

Question 6

Find the largest natural number $k$ such that $k^2$ divides $10!$. Here $10!=10\times9\times8\times\cdots\times2\times1$.

1. 256
2. 360
3. 512
4. 720
5. 5040

Question 7

1. 10
2. 11
3. 12
4. 13
5. 14

Question 8

Khang works Monday through Friday and rests on Saturday and Sunday. In a month with no public holidays, he takes some leave and works only 80% of that month's working days. How many days of leave does he take?

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

Question 9

Seven consecutive integers have the sum of their three smallest numbers equal to 33. What is the sum of their three largest numbers?

1. 37
2. 39
3. 42
4. 45
5. 48

Question 10

1. 43
2. 45
3. 47
4. 49
5. 41

Question 11

1. $(-8)^9$
2. $(-4)^{14}$
3. $(-2)^{26}$
4. $(-16)^6$
5. $(-32)^5$

Question 12

Let $T=123^m+789^n$, where $m,n$ are chosen from the integers 1 through 21. How many different values of $T$ are divisible by 10?

Question 13

1. 515
2. 516
3. 1515
4. 1516
5. 2020

Question 14

Find the units digit of $A=2015^0+2015^1+2015^2+\cdots+2015^{14}+2015^{15}$.

1. 1
2. 5
3. 6
4. 7
5. 9

Question 15

At a gathering of 30 married couples, each person shakes hands with everyone except their own spouse. How many handshakes take place?

1. 1830
2. 1770
3. 1740
4. 1711
5. Other

Question 16

Let $a,b,c,d$ be distinct natural numbers such that $\gcd(a,b,c)=a$, $\gcd(b,c,d)=b$, $\operatorname{lcm}(a,b,c)=45$ and $\operatorname{lcm}(b,c,d)=450$. Find the smallest possible value of $d$.

1. 20
2. 45
3. 50
4. 90
5. 450

Question 17

Two parallel lines each contain four marked points, as shown. How many triangles have their vertices among these eight points?

1. 30
2. 24
3. 36
4. 60
5. 48

Question 18

Question 19

A box contains 400 blue, white, red, yellow and orange marbles. The ratio blue:white:red is $1:2:4$, and the ratio white:yellow:orange is $1:3:6$. What is the least number of marbles that Pi must draw without looking to guarantee 50 marbles of one colour?

1. 50
2. 115
3. 148
4. 196
5. 246

Question 20

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