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VTMO 2017 — Grade 9 — semifinal

Question 1

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 2

The sum of squares of three consecutive odd numbers has four identical digits. Let $S$ be their sum. Find the digit sum of $S$.
  1. $12$
  2. $14$
  3. $15$
  4. $16$
  5. $20$

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

Positive integers $a \lt b \lt 1000$ satisfy $9 a \gt 7 b$. Find the maximum $b - a$.
  1. $210$
  2. $209$
  3. $231$
  4. $198$
  5. $221$

Question 5

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 6

$A = \frac{\sqrt{x} + 1}{\sqrt{x} + 2} - \frac{\sqrt{x} - 2}{\sqrt{x} - 1} - 3 \frac{\sqrt{x}}{x + \sqrt{x} - 2}$. Choose the $x$ interval satisfying $- \frac{1}{3} \lt A \lt - \frac{1}{4}$.
  1. $49 \lt x \lt 100$
  2. $1 \lt x \lt 4$
  3. $5 \lt x \lt 8$
  4. $121 \lt x \lt 130$
  5. $35 \lt x \lt 48$

Question 7

For natural $N \lt 1000$, A rounds $N$ to the nearest hundred and $B$ to the nearest ten. If $A - B = 20$, how many $N$ are possible?
  1. $10$
  2. $20$
  3. $50$
  4. $90$
  5. $100$

Question 8

$512$ people form a line. Move the first to the back and seat the next; repeat until one remains. What was the survivor’s original position?
  1. $1$
  2. $16$
  3. $15$
  4. $64$
  5. $63$

Question 9

Ti climbs $8$ stairs, taking $1 , 2$ or $3$ stairs per step. How many ways?
  1. $24$
  2. $34$
  3. $44$
  4. $55$
  5. $81$

Question 10

$M = n^{2} + n + 37$. How many $n$ in $1 , \ldots , 50$ make $M$ a power of $7$?
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $5$

Question 11

A cube net has faces $12 , 15 , 11 , 16$ in a row, $14$ above $15$ and $13$ below $16$. Find the printed “greatest value of the numbers on three faces sharing a vertex.”

Question illustration

Question 12

Ten pigs are weighed successively. After each new pig the running average increases $1 \,\mathrm{kg}$. How much heavier is pig $10$ than pig $1$?

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

$B = \{ a , a + 1 , \ldots , a + 1050 \}$, where a is natural. Find the smallest a for which $B$ contains exactly $10$ squares.

Question 15

An isosceles trapezoid $ABCD$ with $AB \parallel CD$ is cyclic, $AB = 9 \,\mathrm{cm}$ and $CD = 64 \,\mathrm{cm}$. Find the circumradius.