HKIMO 2019 Grade 5 National Final (Heat) Set 02

A stationery store had a box of pencils. One half but 7 fewer pencils were sold on the first day. One half of the remaining stock but 11 fewer pencils were sold on the second day. Finally 22 pencils remained. How many pencils did the box originally contain?

Source glossary (English–Vietnamese): One half: một nửa; Fewer: ít hơn; Remaining part: phần còn lại; Left: còn lại; Originally: ban đầu.

There are chickens and rabbits. There are 14 more chickens than rabbits and 172 legs in total. How many rabbits are there?

Source glossary (English–Vietnamese): More than: nhiều hơn; In total: tổng số.

Three people take 15 days to complete a task. How many days do five people need for the same task?

Source glossary (English–Vietnamese): Require: yêu cầu; The same: giống nhau.

According to the pattern below, find the missing number.



Source glossary (English–Vietnamese): Pattern: quy luật; Missing number: số còn thiếu.

A bag contains 66 blue balls, 21 red balls and 7 green balls. At least how many balls must be picked to guarantee 5 balls of each color?

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

Find $13+20+27+34+\cdots+83+90+97+104$.

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

Find $\frac1{2\times5}+\frac1{5\times8}+\frac1{8\times11}+\cdots+\frac1{26\times29}+\frac1{29\times32}$.

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

Find

$$\dfrac1{5-\dfrac2{4+\dfrac3{2+\dfrac25}}}.$$

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

Find $\left(\frac53+\frac56+\frac5{12}+\frac5{24}\right)\div\left(\frac92+\frac94+\frac98+\frac9{16}\right)$.

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

Find $\frac1{128}+\frac1{64}+\cdots+\frac14+\frac12+1+2+4+\cdots+64+128$.

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

If the 9-digit number $\overline{20204B22A}$ is divisible by 24, find the maximum value of $2A+B$.

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

The sum of 11 consecutive numbers is 924. Find the smallest number.

Source glossary (English–Vietnamese): Sum: tổng; Consecutive: liên tiếp; Smallest number: số nhỏ nhất.

How many 3-digit numbers are divisible by 7 or by 13?

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

Define $a\otimes b=3ab-(a+1)(b-3)$. Find $5\otimes(2\otimes5)$.

Source glossary (English–Vietnamese): Define: định nghĩa; Operation symbol: ký hiệu phép toán.

A teacher gives pencils to students. Giving 11 to each leaves 4; giving 9 to each leaves 14. How many pencils does the teacher have?

Source glossary (English–Vietnamese): Left: còn lại.

How many rectangles contain both “*” marks in the figure?



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.

A rectangle has area 216 and integer side lengths. Find its smallest possible perimeter.

Source glossary (English–Vietnamese): Area: diện tích; Rectangle: hình chữ nhật; Side: cạnh; Integer: số nguyên; Minimum value: giá trị nhỏ nhất; Perimeter: chu vi.

What is the ratio of an interior angle to an exterior angle of a regular 30-sided polygon?

Source glossary (English–Vietnamese): How many times: gấp bao nhiêu lần; Interior angle: góc trong; Regular 30-sided polygon: đa giác đều 30 cạnh; Exterior angle: góc ngoài.

27 small cubes of side length 1 are combined as shown. Find the surface area.



Source glossary (English–Vietnamese): Cube: khối lập phương; Side length: độ dài cạnh; Pattern: quy luật; Surface area: diện tích toàn phần.