HKIMO 2017 Grade 5 National Final (Heat) Set 04

Question 1

Let $A$ and $B$ be two nonzero digits. The two-digit number $\overline{AB}$ is divisible by 8, and $\overline{BA}$ is the product of three different prime numbers. Find the three-digit number $\overline{ABA}$.

Source glossary (English–Vietnamese): Non-zero digits: các chữ số khác 0; 2-digit numbers: các số có 2 chữ số; Properties: tính chất; Divisible: chia hết; Product: tích; Prime numbers: số nguyên tố.

Question 2

There are chickens and rabbits. The number of legs of the chickens is 26 more than the number of legs of the rabbits. There are 304 legs in total. How many rabbits are there?

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

Question 3

A stationery store had a batch of ballpoint pens. One half and 3 more were sold on the first day. One half of the remaining stock and 9 more were sold on the second day. Finally 54 pens remained. How many pens were there originally?

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

Question 4

John needs 36 days to finish a project. Mary needs 12 days for the same project. If they work together, how many days do they need?

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

Question 5

A bag contains 8 blue balls, 7 red balls and 9 green balls. At least how many must be picked to guarantee 2 balls of each color?

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

Question 6

Find $111111\times1111111$.

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

Question 7

Find $\frac1{3\times4}+\frac1{4\times5}+\frac1{5\times6}+\cdots+\frac1{15\times16}+\frac1{16\times17}$.

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

Question 8

Find $1\times2+2\times3+3\times4+4\times5+5\times6+6\times7+7\times8+8\times9+9\times10$.

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

Question 9

Find $16\times64-28^2$.

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

Question 10

Find $\frac13+\frac19+\frac1{27}+\frac1{81}+\frac1{243}+\frac1{729}$.

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

Question 11

If the 9-digit number $\overline{2018A905B}$ is divisible by 44, find $A+B$.

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

Question 12

Find the number of factors of 480.

Source glossary (English–Vietnamese): Number of factors: số các ước.

Question 13

Define $a\otimes b=ab-b-7$. Find $11\otimes(8\otimes9)$.

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

Question 14

A teacher gives candies to students. Giving 11 candies to each leaves 23; giving 7 to each leaves 91. How many candies does the teacher have?

Source glossary (English–Vietnamese): Left: thừa ra.

Question 15

How many 3-digit numbers are divisible by 3 and have units digit 4?

Source glossary (English–Vietnamese): 3-digit number: số có 3 chữ số; Divisible: chia hết; Unit digit: chữ số hàng đơn vị.

Question 16

How many rectangles are there in the figure below?

Question illustration

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

Question 17

After cutting 3 cm from two neighboring sides of a square card, we obtain a smaller square. Its area is 63 cm² less than the original. Find the perimeter of the original square.

Source glossary (English–Vietnamese): 2 neighboring sides: 2 cạnh kề nhau; Square: hình vuông; Area: diện tích; Less than: ít hơn; Original: ban đầu; Perimeter: chu vi.