HKIMO 2020 Grade 5 National Final (Heat) Set 01

A stationery store had a box of pencils. One half and 7 more pencils were sold on the first day. One half of the remaining stock and 5 fewer pencils were sold on the second day. Finally 28 pencils remained. How many pencils did the box originally contain?

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

According to the pattern below, find the missing number.



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

Six people take 10 days to complete a task. How many days do four people need for the same task?

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

There are chickens and rabbits. There are 24 more chickens than rabbits and 348 legs in total. How many rabbits are there?

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

A bag contains 24 blue balls, 26 red balls and 28 green balls. At least how many balls must be picked to guarantee 8 of each color?

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

Find $12+18+24+30+\cdots+96+102+108+114$.

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

Find $\frac1{3\times7}+\frac1{7\times11}+\frac1{11\times15}+\cdots+\frac1{31\times35}+\frac1{35\times39}$. Write the answer in simplest form.

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

Find $\left(\frac73+\frac79+\frac7{27}+\frac7{81}\right)\div\left(\frac95+\frac9{15}+\frac9{45}+\frac9{135}\right)$. Write the answer in simplest form.

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

Find $\frac1{243}+\frac1{81}+\cdots+\frac19+\frac13+1+3+9+\cdots+81+243$. Write the answer as a mixed fraction.

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

Find $$\dfrac1{5-\dfrac2{4+\dfrac3{3+\dfrac45}}}.$$ Write the answer in simplest form.

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

If the 9-digit number $\overline{20201B16A}$ is divisible by 24, find the maximum value of $A+B$.

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

How many 3-digit numbers are divisible by 7 or by 11?

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

Define $a\otimes b=2ab-(a+1)(b-1)$. Find $4\otimes(3\otimes2)$.

Source glossary (English–Vietnamese): Operation symbol: ký hiệu phép toán.

The sum of 13 consecutive numbers is 1222. Find the largest number.

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

A teacher gives candies to students. Giving 7 to each leaves 18; giving 5 to each leaves 34. How many candies are there?

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

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



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 1260 and integer side lengths. Find the 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.

How many times the exterior angle is the interior angle of a regular 20-sided polygon?

Source glossary (English–Vietnamese): How many times: gấp bao nhiêu lần; Interior angle: góc trong; Regular 20-sided polygon: đa giác đều 20 cạnh; Exterior angle: góc ngoài.

27 small cubes with side length 1 are combined. According to the figure below, 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.