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VTMO 2017 — Grade 8 — semifinal

Question 1

If $x - y \gt x$ and $x + y \gt y$, which conclusion holds?
  1. $x \gt 0 , y \gt 0$
  2. $x \lt 0 , y \lt 0$
  3. $x \lt 0 , y \gt 0$
  4. $x \gt y \gt 0$
  5. $x \gt 0 , y \lt 0$

Question 2

Every $2 \times 2$ block of the $3 \times 3$ grid has the same sum. Its corners are $7 , 5 , 6 , x$ as shown. Find $x$.

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

Question 3

A consists of natural numbers with distinct digits and digit sum $18$. $D$ is its largest minus its smallest member. Find the digit sum of $D$.
  1. $14$
  2. $16$
  3. $18$
  4. $19$
  5. $20$

Question 4

Given $\operatorname{gcd} \left(x , y\right) = 6$ and $x : y = 6 : 7$, find $x + y$.
  1. $24$
  2. $42$
  3. $48$
  4. $78$
  5. $26$

Question 5

Parallelogram $ABCD$ has area $144 \,\mathrm{cm}^{2} , M$ on $AB , E = DM \cap AC , F = CM \cap BD , O = AC \cap BD$. Hatched triangles $ADE$ and $BFC$ total $48 \,\mathrm{cm}^{2}$. Find quadrilateral $MFOE$’s area.

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

Question 6

All $46$ lamps are off. Each move toggles exactly three distinct lamps. What is the minimum moves to turn all on?
  1. $15$
  2. $16$
  3. $17$
  4. $18$
  5. $19$

Question 7

Positive integers $a , b$ below $100$ satisfy $a \lt b$ and $9 a \gt 7 b$. Maximize $b - a$.
  1. $11$
  2. $13$
  3. $17$
  4. $19$
  5. $21$

Question 8

Given $x + \frac{1}{y} = 1$ and $y + \frac{1}{z} = 1$, evaluate $xyz$.
  1. $1$
  2. $- 1$
  3. $- 2$
  4. $2$
  5. $0$

Question 9

A right triangle has shortest side $3 \,\mathrm{cm}$ and smallest angle $15 ^{\circ}$. Find the other leg.
  1. $3 \left(2 + \sqrt{3}\right)$
  2. $6 + \sqrt{3}$
  3. $3 + 3 \sqrt{3}$
  4. $2 + 3 \sqrt{3}$
  5. $2 + \sqrt{3}$

Question 10

Odd numbers are grouped: $\{ 1 \}$; $\{ 3 , 5 \}$; $\{ 7 , 9 , 11 \}$; $\{ 13 , 15 , 17 , 19 \}$; … What is the middle number of group $15$?
  1. $113$
  2. $209$
  3. $211$
  4. $215$
  5. $225$

Question 11

A ticket discount raises attendance $50 \%$ and revenue $20 \%$. All tickets have equal price. What percentage discount was applied?

Question 12

Given $x \ne y , x + 4 y = y^{2}$ and $y + 4 x = x^{2}$, find $x^{2} + y^{2}$.

Question 13

A three-digit code: $923$ has one correct digit in the correct place; $958$ one correct digit wrongly placed; $379$ two correct digits wrongly placed; $142$ none correct; $127$ one correct digit wrongly placed. Find the code.

Question 14

Balls are numbered $1 , \ldots , 50$ and drawn blindly. How many draws guarantee two labels whose product is a square?

Question 15

Square $ABCD$ has side $9 \,\mathrm{cm} , M , N , P , Q$ on $AB , BC , CD , DA$ with $AM = BN = CP = DQ = 3$. Let $E , H$ be $AC$’s intersections with $MQ , NP$, and $F , K BD$’s intersections with $MN , PQ$. Find square $EFHK$’s area.