VTMO 2020 — Grade 4 — vtmo

Question 1

How many digits does $4\times5\times10\times15\times20\times25$ have?

Question 2

Duc and Hai initially have equal card counts. Duc gives Hai $20$ cards, leaving Hai with three times Duc’s remainder. How many did each have originally?

$20$
$40$
$10$
$30$
$60$

Question 3

A mother divides candies equally among Nhi and $11$ friends. If one friend is absent, each receives one more candy. How many did she prepare?

$110$
$120$
$132$
$121$
$12$

Question 4

Two distinct two-digit numbers each have identical digits and are divisible by none of $2 , 3 , 5$. What is their sum?

$33$
$44$
$55$
$66$
$88$

Question 5

A sequence starts $1 , 2$, and each next digit is the last digit of the sum of the previous two, giving $1235831459437 \ldots$ . Which block cannot occur?

$7077$
$3033$
$5050$
$1011$
$9099$

Question 6

$A$ “super even” date has an even day, month, and year (for example $\frac{\frac{14}{02}}{2020}$). How many occur in $2020$?

$180$
$79$
$99$
$90$
$89$

Question 7

Express $12$ as a product of natural numbers whose sum is also $12$. How many unordered ways are there? For example $8 = 1 \times 1 \times 2 \times 4$ and $8 = 1 + 1 + 2 + 4$.

There are $15$ sticks of length $3 \,\mathrm{cm}$ and $10$ of length $2 \,\mathrm{cm}$. What is the fewest additional $1 \,\mathrm{cm}$ sticks needed to use all sticks as the perimeter of an integer-sided rectangle?

Question 9

Starting with $1 + 2 + \ldots + 20$, Jerry repeatedly replaces the sum of two numbers by their difference. Which result is impossible?

$18$
$19$
$20$
$10$
$0$

Question 10

Which cube cannot be folded from the net below?

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Cube $A$
Cube $B$
Cube $C$
Cube $D$
Cube $E$

Question 11

Twenty people count continuously from $1$ clockwise around a circle. Anyone saying a multiple of $2$ or $3$ is eliminated. Continue until one remains. Which number did the winner first say?

Question 12

A square’s one dimension is reduced $10 \,\mathrm{cm}$ and the other increased $15 \,\mathrm{cm}$, making the rectangle shown. How much does its perimeter increase?

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Question 13

Insert plus signs between $8 , 7 , 6 , 5 , 4 , 3 , 2 , 1$ without rearranging the digits. How many ways give sum $90$?

Question 14

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

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Question 15

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 16

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.

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Question 17

A year has two consecutive months totaling $60$ days and starts on Sunday $1$ January. What weekday is its last day? Codes: Sunday $001$, Monday $002$, Tuesday $003$.

Question 18

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

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Question 19

Digits $2 , 0 , 2 , 0 \ldots$ alternate as shown. $S _{n}$ denotes the sum in a chosen $n \times n$ square; the example has $S _{3} = 10$. Find $S _{6}$.

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Question 20

Give four distinct gifts—bear, flowers, robot, book—to Thao, Nhi, Duc, Hai, one each. Hai rejects flowers, Duc rejects book, Nhi rejects robot, Thao rejects bear. How many acceptable assignments exist?