HKIMO 2019 Grade 6 National Final (Heat) Set 02

Find the average of $1994,1998,2002,2006,\ldots,2998,3002,3006$.

Source glossary (English–Vietnamese): Average: trung bình; Sequence: dãy số.

Take Andy’s brother’s age this year, subtract 6, multiply by 5, add 9 and divide by 7. The result is 22. How old is Andy’s brother this year?

Source glossary (English–Vietnamese): Minuses: trừ; Multiplies: nhân; Adds: cộng; Divided by: chia; Result: kết quả.

A bag contains 37 blue balls, 28 red balls and 11 green balls. What is the least number to draw to guarantee 3 non-blue balls and 8 non-green balls?

Source glossary (English–Vietnamese): At least: ít nhất; Ensure: chắc chắn.

There are chickens and rabbits in a cage. The number of chickens is 6 times the number of rabbits. They have 384 legs altogether. How many rabbits are there?

Source glossary (English–Vietnamese): 6 times: 6 lần; In total: tổng cộng.

According to the pattern shown below, find the number at the question mark.

Source glossary (English–Vietnamese): Pattern: quy luật; Question mark: dấu chấm hỏi.

Find $\displaystyle\dfrac{3}{5-\dfrac{1}{3+\dfrac{4}{2-\dfrac25}}}$.

Source glossary (English–Vietnamese): Value: giá trị.

Find $58^2-21\times84$.

Source glossary (English–Vietnamese): Value: giá trị.

Find $\frac1{20}+\frac1{30}+\cdots+\frac1{342}+\frac1{380}$.

Source glossary (English–Vietnamese): Value: giá trị.

Find $\frac1{729}+\frac1{243}+\cdots+\frac19+\frac13+1+3+9+\cdots+243+729$.

Source glossary (English–Vietnamese): Value: giá trị.

Find $5^2+6^2+7^2+8^2+\cdots+14^2+15^2$.

Source glossary (English–Vietnamese): Value: giá trị.

How many simplified fractions with denominator 168 are there?

Source glossary (English–Vietnamese): Simplified fraction: phân số tối giản; Denominator: mẫu số.

The nine-digit number $\overline{20208A21B}$ is divisible by 44. Find the maximum value of $A\times B$.

Source glossary (English–Vietnamese): 9-digit number: số có 9 chữ số; Divisible by: chia hết cho; Maximum value: giá trị lớn nhất.

For $a\ne3$, define $a\otimes b=\frac{3ab}{a-3}+2b$. Find $6\otimes(10\otimes7)$.

Source glossary (English–Vietnamese): Define: định nghĩa; Operation symbol: phép toán; Value: giá trị.

A three-digit number leaves remainder 2 when divided by 5, remainder 4 when divided by 8 and remainder 1 when divided by 13. Find the number.

Source glossary (English–Vietnamese): 3-digit number: số có 3 chữ số; Remainder: số dư; Divided by: chia.

Find the last digit of $9^2+9^3+9^4+9^5+\cdots+9^{98}+9^{99}$.

Source glossary (English–Vietnamese): Last digit: chữ số tận cùng.

How many rectangles in the figure below contain both asterisks?

Source glossary (English–Vietnamese): Rectangle: hình chữ nhật; Figure: hình vẽ.

How many rectangles are there in the figure below?

Source glossary (English–Vietnamese): Rectangle: hình chữ nhật; Figure: hình vẽ.

For a cuboid, increasing its length by 1 cm increases its volume by 32 cm³; increasing its width by 2 cm increases its volume by 76 cm³; increasing its height by 3 cm increases its volume by 57 cm³. Find its original volume in cm³.

Source glossary (English–Vietnamese): Cuboid: hình hộp chữ nhật; Length: chiều dài; Increase: tăng; Volume: thể tích; Width: chiều rộng; Height: chiều cao; Cubic centimeters: xentimét khối.

A square is formed by 324 rectangles measuring 1 cm by 4 cm. Find the decrease in the total perimeter, in cm.

Source glossary (English–Vietnamese): Square: hình vuông; Formed: tạo bởi; Rectangle: hình chữ nhật; Difference: hiệu; Total: tổng cộng; Perimeter: chu vi.

How many times is an interior angle of a regular 18-sided polygon its exterior angle?

Source glossary (English–Vietnamese): Interior angle: góc trong; Regular 18-sided polygon: đa giác đều 18 cạnh; Exterior angle: góc ngoài.