HKIMO 2018 Grade 6 National Final (Heat) Set 03

Find the next term: $2,6,12,20,30,42,\_\_,\ldots$.

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

There are chickens and rabbits in a cage. There are twice as many rabbits as chickens, and 210 legs altogether. How many rabbits are there?

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

Alice is seven times as old as Peter now. In 6 years Alice will be three times as old as Peter. How old is Alice now?

Source glossary (English–Vietnamese): Seven times: 7 lần; Now: hiện nay; Later: sau; Three times: 3 lần.

Alice needs 20 days and Peter 30 days to finish the same project alone. Together with Mary they finish it in 10 days. How many days would Mary need alone?

Source glossary (English–Vietnamese): Together: cùng nhau.

A box has 5 red socks, 9 white socks and 13 blue socks. What is the least number of socks to draw to guarantee four pairs all of the same colour, meaning eight socks of one colour?

Source glossary (English–Vietnamese): At least: ít nhất.

Find $64^2-9\times144$.

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

Find $\frac13+\frac1{15}+\frac1{35}+\cdots+\frac1{399}+\frac1{483}$.

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

Find $\frac12-\frac13-\frac19-\cdots-\frac1{729}-\frac1{2187}$.

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

Find $10^2+11^2+12^2+\cdots+19^2+20^2$.

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

Convert $56/99$ to a decimal.

Source glossary (English–Vietnamese): Convert: đổi; Decimal: số thập phân.

The eleven-digit number $\overline{20190513ABB}$ is divisible by 33, and $A\ne B$. Find the maximum of $A+B$.

Source glossary (English–Vietnamese): 11-digit number: số có 11 chữ số; Divisible by: chia hết; Maximum value: giá trị lớn nhất.

How many simplified fractions with denominator 120 are there?

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

Define $a\oplus b=\frac{ab}{a+1}+b$. Find $4\oplus(2\oplus6)$.

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

A three-digit number leaves remainder 4 on division by 7, remainder 2 on division by 9, and remainder 5 on division by 11. Find the largest such number.

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

The sum of 20 consecutive numbers is 450. Find the second largest number.

Source glossary (English–Vietnamese): Sum: tổng; 20 consecutive numbers: 20 số liên tiếp; Second largest number: số lớn thứ hai.

How many rectangles are in the figure below?

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

A rectangle is cut into 6 equal squares, each of area 16 cm². What is the least increase in the total perimeter, in cm, after cutting?

Source glossary (English–Vietnamese): Rectangle: hình chữ nhật; Equal: bằng nhau; Square: hình vuông; Area: diện tích; At least: ít nhất; Perimeter: chu vi; Increase: tăng lên.

Unit cubes are combined according to the pattern below. Find the total volume of the tenth layer.

Source glossary (English–Vietnamese): Cube: khối lập phương; Side: cạnh; Length: độ dài; Combined: ghép lại; Pattern: quy luật; Total: tổng cộng; Volume: thể tích.

A square is formed from 150 rectangles measuring 2 cm by 3 cm. Find the decrease in total perimeter, in cm.

Source glossary (English–Vietnamese): Square: hình vuông; Rectangle: hình chữ nhật; Decrease: giảm đi; Perimeter: chu vi.

A circle has area 12.56 cm². Its radius equals the diagonal of a square. Find the area of the square, in cm². Take $\pi=3.14$.

Source glossary (English–Vietnamese): Area: diện tích; Circle: hình tròn; Radius: bán kính; Diagonal: đường chéo; Square: hình vuông.