HKIMO 2017 Grade 6 National Final (Heat) Set 04

$A,D$ are nonzero digits. The two-digit numbers satisfy $\overline{DA}=2(\overline{AD}-2)$, and $\overline{AD}$ is a square number. Find $\overline{AD}$.

Source glossary (English–Vietnamese): Two nonzero digits: hai chữ số khác 0; 2-digit numbers: số có hai chữ số; Formed: tạo bởi; Properties: tính chất; Expressed: biểu diễn; Product: tích; Square number: số chính phương.

There are chickens and rabbits in a cage. There are twice as many chickens as rabbits and 352 legs altogether. How many rabbits are there?

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

Andy is three times as old as John now. In 12 years Andy will be twice as old as John. How old is Andy now?

Source glossary (English–Vietnamese): Three times: ba lần; Later: sau; Two times: hai lần; Now: hiện tại.

Amy reads 1 page on day 1, 2 pages on day 2, 3 pages on day 3, and so on. She stops when the total number of pages read is first a multiple of 60. How many pages has she read?

Source glossary (English–Vietnamese): Total: tổng số; Multiple: bội; Until then: đến lúc đó.

There are 6 white, 8 yellow and 11 brown chopsticks mixed together. In the dark, what is the least number to take to guarantee four pairs: three pairs of one colour and one pair of a different colour?

Source glossary (English–Vietnamese): Together: cùng nhau; At least: ít nhất.

Find $222222222\div9$.

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

Find $\frac1{1\times2\times3}+\frac1{2\times3\times4}+\cdots+\frac1{7\times8\times9}+\frac1{8\times9\times10}$.

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

Find $1\times2+2\times3+3\times4+\cdots+20\times21$.

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

Find $144-143-142+141+140-139-138+137+\cdots+4-3-2+1$.

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

Convert $38/99$ to a decimal.

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

The nine-digit number $\overline{2A018901B}$ is divisible by 11. Find $B-A$.

Source glossary (English–Vietnamese): 9-digit number: số có chín chữ số; Divisible: chia hết; Value: giá trị.

How many reduced fractions greater than 0 and less than 1 have denominator 80? The positive-proper restriction is specified in the source glossary.

Source glossary (English–Vietnamese): Simplified fraction: phân số tối giản lớn hơn 0, nhỏ hơn 1; Denominator: mẫu số.

How many three-digit numbers are divisible by 7 or 11, but not by 13?

Source glossary (English–Vietnamese): 3-digit number: số có ba chữ số; Divisible by: chia hết.

A three-digit number leaves remainder 2 on division by 3, remainder 1 on division by 5, and remainder 4 on division by 7. 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 9 consecutive numbers is 126. Find the second largest number.

Source glossary (English–Vietnamese): Sum: tổng; 9 consecutive numbers: 9 số liên tiếp; Second largest number: số lớn thứ hai.

How many rectangles are in the figure below? The original figure continues onto the next source page.

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

A rectangle is cut into 4 equal rectangles. Each small rectangle has area 256 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; Area: diện tích; Perimeter: chu vi; Increase: tăng lên.

Unit cubes are stacked according to the pattern below. If there are 11 layers, find the total surface area.

Source glossary (English–Vietnamese): Cube: khối lập phương; Side: cạnh; Length: độ dài; Combined: ghép lại; Pattern: quy luật; Layer: tầng; Surface area: diện tích toàn phần.

A square is formed from 722 rectangles measuring 1 cm by 2 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 square has side length 18.84 cm. Its perimeter equals the circumference of a circle. Find the circle area, in cm². Take $\pi=3.14$.

Source glossary (English–Vietnamese): Length of side: độ dài cạnh; Square: hình vuông; Perimeter: chu vi; The same: bằng nhau; Circle: hình tròn; Area: diện tích.