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VTMO 2019 — Grade 7 — semifinal

Question 1

Evaluate $\left(1 + 1\right) \left(1 + \frac{1}{2}\right) \left(1 + \frac{1}{3}\right) \ldots \left(1 + \frac{1}{100}\right)$.
  1. $\frac{99}{100}$
  2. $\frac{101}{100}$
  3. $\frac{1}{100}$
  4. $100$
  5. $101$

Question 2

Find the least positive integer $p$ making $540 p$ a perfect square.
  1. $2$
  2. $3$
  3. $5$
  4. $15$
  5. $60$

Question 3

Positive integers satisfy $ab = 32 , bc = 56 , ca = 28$. Find $a + b + c$.
  1. $29$
  2. $28$
  3. $20$
  4. $18$
  5. $19$

Question 4

Dice form a row on a table as shown. Two expose $8$ faces, three $11$. How many faces of thirty dice are visible from all directions?

Question illustration
  1. $110$
  2. $92$
  3. $98$
  4. $90$
  5. $180$

Question 5

Four calendar dates $a , b$; $c , d$ occupy the pictured $2 \times 2$ block. Which equality is always incorrect?

Question illustration
  1. $c - a = d - b$
  2. $c = a + 7$
  3. $d = a + 8$
  4. $a + c = b + d$
  5. $a + d = c + b$

Question 6

An takes one third of the apples and returns $15$. Binh takes one third of the remainder and returns $10$. Their net amounts kept are equal. Chau then takes one third of the remainder and returns $x$ to keep the same amount. Find $x$.
  1. $0$
  2. $5$
  3. $8$
  4. $10$
  5. $2$

Question 7

How many three-digit numbers have only even digits?
  1. $100$
  2. $48$
  3. $125$
  4. $120$
  5. $450$

Question 8

Find the quadrilateral area for ordered vertices $\left(3 , 3\right) , \left(- 2 , 1\right) , \left(- 1 , - 3\right) , \left(4 , - 1\right)$.
  1. $22$
  2. $23$
  3. $27$
  4. $30$
  5. $32$

Question 9

$ABC$ is right at $A$, altitude $AH = 1 \,\mathrm{cm}$, and angle $B = 60 ^{\circ}$. Find $BC$.
  1. $\frac{2}{\sqrt{3}}$
  2. $2 \sqrt{2}$
  3. $2 \sqrt{3}$
  4. $\frac{4}{\sqrt{3}}$
  5. $2 \frac{\sqrt{2}}{\sqrt{3}}$

Question 10

How many unordered positive-integer pairs satisfy $x + y + xy = 63$?
  1. $3$
  2. $2$
  3. $4$
  4. $6$
  5. $1$

Question 11

Fifty coins start heads-up. Flip $25$, then $29$, then $14$ coins; finally all are tails-up. How many are flipped three times?
  1. $9$
  2. $11$
  3. $15$
  4. $22$
  5. $33$

Question 12

$AB = BD = BC$ in the pictured quadrilateral, and $DBC = 76 ^{\circ}$. Find $DAC$.

Question illustration
  1. $39$
  2. $32$
  3. $40$
  4. $35$
  5. $38$

Question 13

How many three-digit numbers have adjacent digits differing by $1$, such as $123 , 434 , 987$?
  1. $32$
  2. $31$
  3. $30$
  4. $33$
  5. $35$

Question 14

$AD$ bisects $A , D$ lies on $BC$, and $ADB : ADC = 4 : 5$. The perpendicular bisector of $BC$ meets $AC$ at $M$. Find $ABM$.
  1. $20$
  2. $30$
  3. $40$
  4. $50$
  5. $60$

Question 15

Starting with seven entries $1 , \ldots , 7$, each move decreases one entry by $1$ and increases another by $5$. The final entries are $2012 , \ldots , 2017 , x$. Which listed $x$ is possible?
  1. $2018$
  2. $2017$
  3. $2019$
  4. $2020$
  5. $2021$

Question 16

How many four-digit numbers with distinct digits chosen from $1 , 2 , 3 , 4 , 5$ are divisible by $6$?

Question 17

A worker paints four walls and the ceiling of a room in $20$ hours. Doubling its length and width while keeping height unchanged requires $50$ hours at the same rate. How long does the original ceiling take?

Question 18

$A , B , C , D$ have $76$ candies. Simultaneously they retain $1 , 2 , 3 , 4$ respectively, and share the rest equally among the other three. Final amounts are equal. Find $A$’s initial amount.

Question 19

An, Binh, Chau run $200 m$ at constant speeds, An finishing first. After $25$ seconds their combined distance is $500 m$. When An finishes, the others’ remaining distances sum $40 m$. Find An’s time.

Question 20

$ABCD$ is a square of side $5 \,\mathrm{cm}$. Interior point $M$ has $MA = 3 \,\mathrm{cm}$ and $MA$ perpendicular to $MB$. Find $MD^{2} + MC^{2}$.

Question illustration