Skip to content / Đến nội dung
ThichHocToan.com

VTMO 2017 — Grade 7 — semifinal

Question 1

$P$ is the product of the first $2017$ primes. Find its last digit.
  1. $0$
  2. $1$
  3. $5$
  4. $7$
  5. Other

Question 2

$M$ is the least composite number not divisible by any integer $2$ through $50$. Find its digit sum.
  1. $12$
  2. $14$
  3. $18$
  4. $19$
  5. $20$

Question 3

$AB : AC : BC = 3 : 4 : 5$. Find the ratio of altitudes from $C , B , A$.
  1. $5 : 4 : 3$
  2. $3 : 4 : 5$
  3. $20 : 15 : 12$
  4. $12 : 15 : 20$
  5. $9 : 8 : 6$

Question 4

A consists of natural numbers with distinct digits summing $18$. Let $D$ be the maximum minus the minimum. Find its digit sum.
  1. $12$
  2. $9$
  3. $18$
  4. $15$
  5. Other

Question 5

Distinct natural $x , y$ have $\operatorname{gcd} 6$ and ratio $6 : 7$. Find their sum.
  1. $18$
  2. $42$
  3. $48$
  4. $78$
  5. Other

Question 6

$A = 3 \times 3$ grid has corner entries $7 , 5 , 6 , x$ as shown. All $2 \times 2$ subgrids have equal sums. Find $x$.

Question illustration
  1. $2$
  2. $3$
  3. $6$
  4. $4$
  5. $8$

Question 7

Parallelogram $ABCD$ has area $144 \,\mathrm{cm}^{2} , M$ lies on $AB$, and the hatched regions total $48 \,\mathrm{cm}^{2}$. Find area $MFOE$.

Question illustration
  1. $36$
  2. $18$
  3. $12$
  4. $9$
  5. $24$

Question 8

Concatenate squares $1 , 4 , 9 , 16 , 25 , 36 , 49 , \ldots$ Find digit $100$.
  1. $1$
  2. $4$
  3. $6$
  4. $9$
  5. $8$

Question 9

Five nonparallel lines have exactly one triple concurrence and otherwise distinct intersection points. How many plane regions result?
  1. $15$
  2. $16$
  3. $17$
  4. $18$
  5. $20$

Question 10

Teams $A , B , C$ produce equal quantities with equal individual productivity, in $12 , 15 , 21$ days. $A$ has $30$ more workers than $C$. How many more than $B$?
  1. $10$
  2. $11$
  3. $12$
  4. $13$
  5. $14$

Question 11

A park reduces ticket prices; visitors rise $50 \%$ and revenue rises $20 \%$. What percentage price reduction occurred?

Question 12

A four-digit code: $2896$ has exactly one correct digit in its correct position; $5273$ has exactly two correct digits, both misplaced; $1702$ has all four correct but all misplaced. Find it.

Question 13

Triangle $ABC$ has $B = 40 ^{\circ} , C = 80 ^{\circ}$. $MN$ is parallel to $AB$ with $M$ on $CA , N$ on $CB$, and $AM = MN$. Find angle $BAN$.

Question 14

One traveler takes $3$ hours at speed $v$. Another uses speeds $2 v$ and $\frac{v}{3}$ and takes $5$ hours. How many minutes were spent at $2 v$?

Question 15

$ABC$ is right at $A , AB = 3 \,\mathrm{cm} , AC = 4 \,\mathrm{cm}$. The angle $- B$ bisector meets $AC$ at $D$. Find $2 AD$.