VTMO 2026 - Grade 5 - Final - SET 7 - Self-study

A rectangular fish tank has base area $48\text{ dm}^2$ and water depth $6$ dm. Taps A and B are opened together: A pours in $32$ litres per minute, while B drains $18$ litres per minute. Both taps are closed after 36 minutes. The water depth is then $\frac34$ of the tank's height. What is the tank's height in decimetres? Use $1$ litre $=1\text{ dm}^3$.

Exactly one of four dwarfs A, B, C and D stole the king's gold. Each makes two statements:
• A: “B is a liar.” “C or D stole the gold.”
• B: “C is a liar.” “D or A stole the gold.”
• C: “D is a liar.” “A or B stole the gold.”
• D: “A is a liar.” “B or C stole the gold.”
A liar makes at least one false statement; a truthful dwarf makes two true statements. How many of the eight statements are true?

Some children go mushroom picking. If An gives Bình half of An's mushrooms, all the children will have equal numbers of mushrooms. If instead An gives Châu all of An's mushrooms, Châu will have as many mushrooms as all the other children together. How many children are there? Show your reasoning.

Practice clarification: The children have collected a positive total number of mushrooms.

A father walks to school with his two children, Pi and Ksi. While the father takes 5 steps, Ksi takes 4 steps. While Ksi takes 5 steps, Pi takes 4 steps. Ksi and Pi take 360 steps altogether. How many steps does the father take?

A teacher has a bag of candies. She gives An $\frac13$ of the candies she has, then gives Bình $\frac15$ of the remaining candies. Next she gives Châu as many candies as An and Bình received altogether. Finally, she gives Dũng all 10 candies left. How many candies did she have initially?

A natural number is formed by writing the integers from 6 to 100 consecutively:
$$678910111213\ldots9899100.$$
What is its middle digit?

Place each integer from 1 to 13 exactly once in the small circles. The difference between any two adjacent numbers (larger minus smaller) must be less than 3. The positions of 6, $x$ and $y$ are shown. Find $x+y$.

The calendar shown is February 2024. The Tuesday dates in that month sum to $6+13+20+27=66$. What is the smallest possible sum of the day-of-month numbers of any five consecutive Tuesdays? The five Tuesdays may cross a month boundary.

The IQ mathematics classes at Titan have 280 students, of whom 40% are girls. How many girls must join, with no other changes, for girls to make up 60% of all students?

Titan’s Grade 5 thinking-mathematics class IQ5 has these properties: no two girls were born on the same day of the week, and no two boys were born in the same month. Adding any one more boy or any one more girl would make the corresponding property fail. How many students are in the class?

If 162 sweets are shared equally among the students of class 5A, 6 sweets remain. If 375 sweets are shared equally, 6 more sweets are needed to complete equal shares. If 273 sweets are shared equally, none remain. What is the greatest possible number of students?

Data check: the quantities are retained as given. If they are inconsistent, answer “No solution”.

Minh has five boxes: one red, two blue and two yellow. He stacks all five with these rules:
• Adjacent boxes have different colours.
• The top box is not yellow.
• The middle (third) box is not red.
• A yellow box is immediately below the red box.
• The bottom box is not yellow.
Give the colours in order from bottom to top.

Fill the circles with natural numbers. Each number below a pair is the difference of the two numbers immediately above it (larger minus smaller). Find the sum of all possible numbers that can replace “?”.