HKIMO 2020 Grade 6 National Final (Heat) Set 01

According to the pattern below, find the number in the blank.

Source glossary (English–Vietnamese): Pattern: quy luật; Number: số; Blank: chỗ trống.

Take Alice’s brother’s age this year, subtract 5, multiply by 6, add 6, and divide by 9. The result is 24. How old is he this year?

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

A bag contains 35 blue,29 red and 13 green balls. What is the least number to draw to guarantee 5 non-blue balls and 10 non-green balls?

Source glossary (English–Vietnamese): At least: ít nhất; How many: có bao nhiêu; Ensure: chắc chắn.

Find the average of $1986,1992,1998,2004,\ldots,2220,2226,2232$.

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

There are chickens and rabbits in a cage; the number of chickens is 8 times the number of rabbits. They have 480 legs altogether. How many rabbits are there?

Source glossary (English–Vietnamese): 8 times: 8 lần; Total: tổng cộng.

Find $\dfrac 4{5-\dfrac 3{3+\dfrac 2{1-\dfrac 15}}}$. Write the answer in simplest form.

Source glossary (English–Vietnamese): Value: giá trị; Simplest form: dạng tối giản.

Find $61^2-13\times 117$.

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

Find $\frac 1{256}+\frac 1{128}+\frac 1{64}+\cdots+\frac 14+\frac 12+1+2+4+\cdots+64+128+256$. Write the answer as a mixed fraction.

Source glossary (English–Vietnamese): Value: giá trị; Mixed fraction: hỗn số.

Find $\frac 1{28}+\frac 1{70}+\frac 1{130}+\cdots+\frac 1{418}+\frac 1{550}$. Write the answer in simplest form.

Source glossary (English–Vietnamese): Value: giá trị; Simplest form: dạng tối giản.

Find $7^2+8^2+9^2+\cdots+17^2+18^2+19^2$.

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

How many simplified fractions with denominator 252 are there?

Source glossary (English–Vietnamese): Simplified fraction: phân số tối giản dương và nhỏ hơn 1; Denominator: mẫu số.

The nine-digit number $\overline{20202 A 12 B}$ is divisible by 44. Find the maximum 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 $a\ne 3$, define $a\otimes b=\frac{3 ab}{a-3}$. Find $16\otimes(13\otimes 10)$.

Source glossary (English–Vietnamese): Operation symbol: kí hiệu phép toán; Value: giá trị.

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

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

A three-digit number leaves remainder 2 on division by 3, remainder 1 on division by 7 and remainder 4 on division by 8. Find the largest such number.

Source glossary (English–Vietnamese): 3-digit number: số có 3 chữ số; Remainder: số dư; Divide by: chia cho; Largest: lớn nhất.

How many rectangles contain both asterisks in the figure below?

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

How many rectangles are 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 volume by $24\ \mathrm{cm}^3$; increasing its width by 2 cm increases volume by $56\ \mathrm{cm}^3$; increasing its height by 3 cm increases volume by $126\ \mathrm{cm}^3$. Find the original volume in cubic centimetres.

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

How many times as large as an exterior angle is an interior angle of a regular 12-sided polygon?

Source glossary (English–Vietnamese): How many times: gấp bao nhiêu lần; Interior angle: góc trong; Regular 12-sided polygon: đa giác đều 12 cạnh; Exterior angle: góc ngoài.

A square is formed from 243 rectangles measuring 1 cm by 3 cm. Find the difference between the total perimeters before and after joining, in cm.

Source glossary (English–Vietnamese): Square: hình vuông; Rectangle: hình chữ nhật; Difference: hiệu; Total perimeter: tổng chu vi.