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VTMO 2016 — Grade 7 — semifinal

Question 1

Subtract the sum of the first $50$ positive even integers from the sum of the first $51$ positive odd integers.
  1. $50$
  2. $51$
  3. $100$
  4. $102$
  5. $- 51$

Question 2

Two positive integers sum to $1323$ and have $\operatorname{gcd} 147$. How many unordered pairs are possible?
  1. $3$
  2. $4$
  3. $5$
  4. $9$
  5. $10$

Question 3

What is the smaller angle between the hands at $4 : 20$?
  1. $5$
  2. $6$
  3. $8$
  4. $10$
  5. $20$

Question 4

Use $2 , 3 , 4 , 5 , 6 , 7 , 8$ once to minimize $UVW - \left(XY + MN\right)$.
  1. $70$
  2. $73$
  3. $82$
  4. $172$
  5. $105$

Question 5

$AB = 5 \,\mathrm{cm} , BC = 8 \,\mathrm{cm}$, angle $ABC = 30 ^{\circ}$. Find the triangle area.
  1. $8$
  2. $10$
  3. $12$
  4. $14$
  5. $16$

Question 6

How many natural numbers $0$ through $30$ are differences of cubes of two natural numbers?
  1. $4$
  2. $5$
  3. $6$
  4. $7$
  5. $8$

Question 7

A car must average $48 \frac{\,\mathrm{km}}{h}$ over its trip. The first half-distance is traveled at $40 \frac{\,\mathrm{km}}{h}$. What minimum speed is needed for the second half?
  1. $50$
  2. $55$
  3. $60$
  4. $65$
  5. $70$

Question 8

Find the last digit of $1 \times 3 \times 5 \times 7 + 3 \times 5 \times 7 \times 9 + \ldots + 2011 \times 2013 \times 2015 \times 2017$.
  1. $0$
  2. $2$
  3. $4$
  4. $6$
  5. $8$

Question 9

The two diagonals of $a = 3 \times 3$ square pass through five small cells, as shown. How many cells do the two diagonals of an $11 \times 11$ grid pass through?

Question illustration
  1. $19$
  2. $20$
  3. $21$
  4. $22$
  5. $23$

Question 10

On $a = 100 m$ square, An and Binh start at opposite vertices, both counterclockwise, at $50$ and $40 m /$ min. An rests one minute at each corner, Binh two. When does An first catch Binh?
  1. $18$
  2. $17$
  3. $19$
  4. $16$
  5. $20$

Question 11

How many three-digit numbers have digit product $36$? Examples $229 , 419$.

Question 12

Twenty girls form a line, and four boys enter with $17 , 14 , 5 , 2$ girls before them. Each girl counts boys before her. Find the sum of the girls’ counts.

Question 13

Among $65$ pupils wearing white or blue, there are $25$ boys in white, $23$ pupils in blue, and $28$ girls. How many girls wear blue?

Question 14

A cube net shows $1 , 5 , 6 , x , y , z$ as pictured. Both $5$ and $x$ equal the mean of their four adjacent faces. Find $x$.

Question illustration

Question 15

Rectangle $ADEF$ lies in right triangle $ABC$ as shown. $BD = 10 \,\mathrm{cm} , FC = 15 \,\mathrm{cm}$. Find the rectangle area.

Question illustration