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VTMO 2021 — Grade 6 — vtmo

Question 1

Evaluate $1 \times 2 + 3 - 4 \times \left(5 - 6\right) \times 7 \times 8 \times 9$.
  1. $2017$
  2. $2018$
  3. $2019$
  4. $2020$
  5. $2021$

Question 2

For natural $a , b , c , d , 2^{12} = 4^{a} = 8^{b} = 16^{c} = 64^{d}$. Find their sum.
  1. $10$
  2. $12$
  3. $13$
  4. $15$
  5. $16$

Question 3

Fill $4 , 5 , 6 , 7 , 8 , 9$ once around the pictured ring, with $7$ given. Adjacent sums must be prime. Which number occupies“?” opposite $7$?

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  1. $9$
  2. $8$
  3. $6$
  4. $5$
  5. $4$

Question 4

In the pictured $4 \times 4$ grid, how many rectangles, including squares, contain the diamond but exclude the black square?

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  1. $32$
  2. $8$
  3. $24$
  4. $34$
  5. Other

Question 5

Find the greatest $n$ such that $2^{n}$ divides $\operatorname{lcm} \left(1 , 2 , \ldots , 2020\right)$.
  1. $10$
  2. $12$
  3. $1024$
  4. $1010$
  5. $2020$

Question 6

How many natural numbers below $2021$ have exactly three positive divisors?
  1. $10$
  2. $14$
  3. $18$
  4. $25$
  5. Other

Question 7

Tom tells truth only on dates divisible by $3$, Jerry only on dates divisible by $4$. How many dates in January $2021$ allow both to say“I lied yesterday”?
  1. $12$
  2. $10$
  3. $8$
  4. $9$
  5. Other

Question 8

A consists of repunits of lengths $1$ through $36$. How many divide the $36$-digit repunit?
  1. $6$
  2. $8$
  3. $9$
  4. $10$
  5. $12$

Question 9

All lights in $a = 4 \times 4$ grid are on. Touching a light toggles it and its side-adjacent neighbors. What minimum touches turn them all off?

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  1. $3$
  2. $4$
  3. $5$
  4. $6$
  5. $7$

Question 10

$n$! is the product $1$ through $n$. Find the least $n$ such that $n$! is divisible by both $2020$ and $2021$.
  1. $2021$
  2. $43$
  3. $47$
  4. $101$
  5. $2020$

Question 11

An odd three-digit $N$ has distinct digits and equals $25$ times its digit sum. Find $N$.

Question 12

A unit-cube solid has the front and right projections shown. What maximum total surface area is possible?

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Question 13

Chau and her mother’s ages total $50$. In $17$ years, Chau will be as old as her mother was at Chau’s birth. Find Chau’s present age.

Question 14

A machine halves even natural numbers and subtracts $1$ from odd ones until $0$. What least number needs exactly ten steps? For example $6$→$3$→$2$→$1$→$0$ takes four.

Question 15

An Egyptian fraction is $\frac{1}{n}$ with $n \ge 2$. Write $\frac{4}{15} = \frac{1}{a} + \frac{1}{b}$. Minimize $a + b$.

Question 16

Rectangle $ABCD$ is split into $AEFD$ and $BCFE$ as pictured. $BCFE$ has twice the perimeter and four times the area of $AEFD$. $AD = 10 \,\mathrm{cm}$. Find $AB$.

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Question 17

Two squares are overlaid as pictured, the larger side $8 \,\mathrm{cm}$. Striped area exceeds dotted area by $4 \,\mathrm{cm}^{2}$. Find the total unshaded area.

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Question 18

Within $2001 \text{--} 2100$, what maximum consecutive-year run has January $13$ not on a Friday?

Question 19

Distinct letters represent distinct digits, and $VIETNAM + TITAN = MYSTERY , E \lt 3$. Find $T + I + T + A + N$.

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Question 20

Student $i$ receives $i$ candies then one twenty-first of the remainder. The last receives all remaining candies, exactly his/her index. All receive equal amounts. How many students are there?