HKIMO 2020 Heat P5 [PDF]

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HONG KONG INTERNATIONAL MATHEMATICAL OLYMPIAD HEAT ROUND 2020

Primary 5 Time allowed:

90

minutes

Question Paper Instructions to Contestants: 1.

Each contestant should have ONE Question-Answer Book which CANNOT be taken away.

2.

There are 5 exam areas and 5 questions in each exam area. There are a total of 25 questions in this Question-Answer Book. Each carries 4 marks. Total score is 100 marks. No points are deducted for incorrect answers.

3.

All answers should be written on ANSWER SHEET.

4.

NO calculators can be used during the contest.

5.

All figures in the paper are not necessarily drawn to scale.

6.

This Question-Answer Book will be collected at the end of the contest.

THIS Question-Answer Book CANNOT BE TAKEN AWAY.

DO NOT turn over this Question-Answer Book without approval of the examiner. Otherwise, contestant may be DISQUALIFIED.

Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

All answers should be written on the ANSWER SHEET.

Open-Ended Questions (1st ~25th) (4 points for correct answer, no penalty point for wrong answer)

Logical Thinking 1.

A stationary store had a box of pencils. One half but 7 fewer pencils were sold on the first day. One half of the remaining part and 11 fewer pencils were sold on the second day. Finally 22 pencils are left. How many pencil(s) did the box contain originally? Sebuah toko alat tulis mempunyai sekotak pensil. Separoh kurang 7 pensil terjual pada hari pertama. Separoh dari sisanya kurang 11 pensil terjual pada hari kedua. Akhirnya 22 pensil tersisa. Berapa banyak pensil yang ada pada kotak tersebut pada awalnya?

2.

There are some chickens and rabbits. The number of chickens is 14 more than the number of rabbits. If there are 172 legs in total, how many rabbit(s) is / are there? Terdapat sejumlah ayam dan kelinci. Jumlah ayam 14 lebih banyak dari jumlah kelinci. Jika terdapat keseluruhan 172 kaki, berapa banyak kelinci ada di sana?

3.

It requires 3 people to take 15 days to complete a task. How many day(s) is / are needed for 5 people to finish the same task? Diperlukan 3 orang untuk menyelesaikan sebuah pekerjaan dalam 15 hari. Berapa hari yang diperlukan 5 orang untuk menyelesaikan pekerjaan yang sama?

4.

According to the pattern shown below, what is the number in the blank? Berdasarkan pola di bawah ini, berapa bilangan yang diwakili ‘?’?

24

6

7

49

9

?

20

5

3

21

1

5

Question 4 Soal no. 4 5.

There are 66 blue balls, 21 red balls and 7 green balls in a bag. At least how many balls should be picked up to ensure there are 5 balls for each colour? Terdapat 66 bola biru, 21 bola merah dan 7 bola hijau di dalam sebuah kantong. Setidaknya berapa bola yang harus diambil untuk memastikan ada terambil 5 bola dari masing-masing warna? Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

All answers should be written on the ANSWER SHEET.

Arithmetic

6.

Find the value of 13 + 20 + 27 + 34 + ... + 83 + 90 + 97 + 104 . Carilah nilai dari 13 + 20 + 27 + 34 + ... + 83 + 90 + 97 + 104 .

7.

Find the value of

8.

Find the value of

1 1 1 1 1 . + + + ... + + 2  5 5  8 8 11 26  29 29  32 1 1 1 1 1 Carilah nilai dari . + + + ... + + 2  5 5  8 8 11 26  29 29  32

1

.

2

5−

3

4+

2+

1

Carilah nilai dari

.

2

5− 4+

3 2+

9.

2 5

2 5

5 5 5 5  9 9 9 9  Find the value of  + + +    + + +  .  3 6 12 24   2 4 8 16  5 5 5 5  9 9 9 9  Carilah nilai dari  + + +    + + +  .  3 6 12 24   2 4 8 16 

10. Find the value of

1 1 1 1 + + ... + + + 1 + 2 + 4 + ... + 64 + 128 . 128 64 4 2

Carilah nilai dari

1 1 1 1 + + ... + + + 1 + 2 + 4 + ... + 64 + 128 . 128 64 4 2

Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

All answers should be written on the ANSWER SHEET.

Number Theory

11. If a 9-digit number 20204B22 A is divisible by 24, find the maximum value of 2A + B . Jika sebuah bilangan 9-angka 20204B22 A dapat dibagi 24, carilah nilai maksimum dari 2 A + B .

12. The sum of 11 consecutive numbers is 924. Find the value of the smallest number. Hasil penjumlahan dari 11 bilangan berurutan adalah 924. Carilah nilai dari bilangan terkecil.

13. How many 3-digit number(s) either divisible by 7 or divisible by 13 is / are there? Berapa banyak bilangan 3-angka yang dapat dibagi 7 atau dapat dibagi 13? 14. Define the operation symbol a  b = 3  a  b − ( a + 1)  ( b − 3) , find the value of 5  ( 2  5 ) . Jika didefinisikan a  b = 3  a  b − ( a + 1)  ( b − 3) , carilah nilai dari 5  ( 2  5 ) . 15. A teacher gives some pencils to students. If everyone gets 11 pencils, 4 pencils will be left. If everyone gets 9 pencils, 14 pencils will be left. How many pencil(s) does the teacher have? Seorang guru memberikan pensil pada murid-muridnya. Jika semua mendapat masing-masing 11 pensil, 4 pensil akan tersisa. Jika semua mendapat masing-masing 9 pensil, 14 pensil akan tersisa. Berapa banyak pensil yang dipunyai si guru?

Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

All answers should be written on the ANSWER SHEET.

Geometry

16. How many rectangle(s) with 2 “*” is / are there in the figure below? Berapa banyak persegipanjang yang mengandung 2 “*” yang ada pada gambar di bawah ini?

* *

Question 16 Soal no. 16

17. How many rectangle(s) is / are there in the figure below? Berapa banyak persegipanjang yang ada pada gambar di bawah ini?

Question 17 Soal no. 17

Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

All answers should be written on the ANSWER SHEET.

18. The area of a rectangle is 216. If sides of the rectangle are integers, find the minimum value of the perimeter of the rectangle. Luas dari sebuah persegipanjang adalah 216. Jika panjang sisi dari persegipanjang tersebut bulat, carilah nilai minimum dari keliling persegipanjang tersebut.

19. How many time(s) is the interior angle of a regular 30-sided polygon as the exterior angle? Berapa kali lebih besar sudut interior (dalam) dari sudut eksterior (luar) nya dalam sebuah segi-30 beraturan?

20. 18 small cubes with side length 1 are combined. According to the pattern shown below, find the surface area. 18 kubus kecil dengan panjang sisi 1 disusun. Pada gambar di bawah ini, carilah luas permukaannya.

Question 20 Soal no. 20

Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

All answers should be written on the ANSWER SHEET.

Combinatorics

21. 17 students join a competition. Only top 3 students can be promoted to next round. How many different way(s) of promotion advance is / are there? 17 pelajar mengikuti sebuah kompetisi. Hanya 3 peserta teratas akan lolos ke babak berikutnya. Berapa banyak kombinasi 3 peserta lolos yang ada?

22. A flight of stairs has 9 steps. Peter can go up for 1 step, 2 steps or 3 steps each time. How many way(s) is / are there for Peter to go up the stairs? Sebuah tangga mempunyai 9 anak tangga. Peter dapat menaiki 1, 2, atau 3 anak tangga pada tiap langkahnya. Berapa banyak cara Peter dapat menaiki tangga tersebut?

23. It is known that there are 7 classes of grade 5 in a primary school. At least how many of 366 grade 5 students that are in the same class? Diketahui terdapat 7 kelas pada kelas 5 Sekolah Dasar. Setidaknya ada berapa dari 366 murid kelas 5 Sekolah Dasar ada pada kelas yang sama?

24. Numbers are drawn from 90 integers 10 to 99. At least how many number(s) is / are drawn at random to ensure that there are two numbers whose sum is 75? Bilangan diambil secara acak dari 90 bilangan dari 10 sampai dengan 99. Setidaknya berapa banyak bilangan harus diambil untuk memastikan ada dua bilangan yang hasil penjumlahan keduanya adalah 75? 25. Choose 3 digits from 1, 2, 3, 4, 6, 8 and 0 to form 3-digit numbers. How many even number(s) greater than 400 is / are there? (The repetition of digits is not allowed) Pilihlah 3 angka dari 1, 2, 3, 4, 6, 8 dan 0 untuk membentuk bilangan 3-angka. Berapa banyak bilangan genap yang lebih besar dari 400 dapat dibentuk? (Pengulangan angka tidak diperbolehkan)

~ End of Paper ~

Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.