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VTMO 2020 — Grade 5 — vtmo

Question 1

Find the sum of the last six digits of $1 \times 2 \times 3 \times \ldots \times 25$.
  1. $0$
  2. $12$
  3. $16$
  4. $20$
  5. $10$

Question 2

Khoa earns $3$ per $2$ hours, works $6$ hours daily, $6$ days weekly, for $6$ weeks. Find total earnings.
  1. $432$
  2. $108$
  3. $296$
  4. $324$
  5. $1296$

Question 3

There are $20$ books. An read $19$, Binh $12$, Chau $11$. What minimum number were read by all three?
  1. $10$
  2. $12$
  3. $1$
  4. $11$
  5. $2$

Question 4

Starting with $1 + 2 + \ldots + 20$, Jerry repeatedly replaces the sum of two numbers by their difference. Which result is impossible?
  1. $18$
  2. $19$
  3. $20$
  4. $10$
  5. $0$

Question 5

Write $20$ as a product of natural numbers whose sum is also $20$. Ignoring order, how many ways are there? For example $8 = 1 \times 1 \times 2 \times 4$ and $8 = 1 + 1 + 2 + 4$.
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $0$

Question 6

There are twenty $3 \,\mathrm{cm}$ sticks and seven $2 \,\mathrm{cm}$ sticks. What minimum number of $1 \,\mathrm{cm}$ sticks must be added to use all sticks as an integer-sided rectangle boundary?
  1. $0$
  2. $1$
  3. $2$
  4. $3$
  5. $4$

Question 7

Two joined pipes measure $355 \,\mathrm{cm}$; three identical joined pipes measure $530 \,\mathrm{cm}$ as pictured. Find one pipe’s length.

Question illustration
  1. $180 \,\mathrm{cm}$
  2. $175 \,\mathrm{cm}$
  3. $170 \,\mathrm{cm}$
  4. $165 \,\mathrm{cm}$
  5. $160 \,\mathrm{cm}$

Question 8

Alternating $2 s$ and $0 s$ fill rows and columns as shown. $S _{n}$ is the sum in any $n \times n$ square; the illustrated $S _{3} = 10$. Find $S _{10}$.

Question illustration
  1. $102$
  2. $92$
  3. $96$
  4. $104$
  5. $100$

Question 9

Append a digit to a natural number $N$. The new number exceeds $N$ by $2020$. Find the appended digit.
  1. $2$
  2. $4$
  3. $6$
  4. $8$
  5. $0$

Question 10

The figure consists of rectangles twice as long as wide. How many single-rectangle removals leave the perimeter unchanged?

Question illustration
  1. $2$
  2. $4$
  3. $1$
  4. $3$
  5. $0$

Question 11

Blue and red candy bags initially contain equal amounts. Each pupil chooses $2$ blue $+ 2$ red or $1$ blue $+ 3$ red. Only $20$ blue remain. How many chose option $2$?

Question 12

A digital clock displays $HH : MM$ from $00 : 00$ to $23 : 59$. How many daily displays contain exactly two $2 s$ and two $0 s$?

Question 13

$A = 14 \,\mathrm{cm}$ square contains an inscribed circle, which contains an inscribed smaller square as pictured. Find the shaded area outside the smaller square.

Question illustration

Question 14

Write eight consecutive natural numbers on a cube’s eight vertices. What maximum number of edges have an odd sum of endpoints?

Question illustration

Question 15

Buying a watch at $40 \%$ off leaves mother $16$; at full price she would be short $4$. Find its full price.

Question 16

Distinct letters are distinct digits. Maximize $\overline{VIETNAM}+\overline{TITAN}+\overline{MATH}$. Find $T + I + T + A + N$.

Question illustration

Question 17

Concatenate consecutive odd numbers $1 , 3 , 5 , 7 , 9 , 11 , \ldots$ until two $9 s$ appear consecutively. What minimum number of terms have been written?

Question 18

Place $1$ through $6$ once on the cube’s faces. Each vertex value is the sum of its three incident faces. Find the sum of all eight vertex values.

Question 19

An empty $2020$-liter tank is filled at $20$ liters/minute. After $20$ minutes it starts leaking $2$ liters/minute while pumping continues. How long from the start until full?

Question 20

Fill $a = 4 \times 5$ grid with $0 , 1 , 2$ so each row and column sum is divisible by $3$. What maximum number of $1 s$ is possible?