# WMI Preliminary Round 2021 [Grade 6B]

World Mathematics Invitational (WMI) is the first international competition founded by Taiwan. It gathers institutes and organizations worldwide that make efforts in promoting and popularizing mathematics. Through interacting with other math-loving students that represent their countries, students can expand their worldview, experience different cultures, and thus their horizon as well as their future will be broaden. (sc : http://www.wminv.org/)

Berikut ini soal dan solusi WMI grade 6B tahun 2021

1.$$\frac{1}{7}×\frac{3}{4}+\frac{2}{7}×\frac{1}{6}+\frac{6}{7}×\frac{1}{12}=…$$

(A) $$\frac{23}{84}$$
(B) $$\frac{20}{84}$$
(C) $$\frac{19}{84}$$
(D) $$\frac{17}{84}$$

$$\frac{1}{7}×\frac{3}{4}+\frac{2}{7}×\frac{1}{6}+\frac{6}{7}×\frac{1}{12}$$
$$=\frac{1}{7}\left(\frac{3}{4}+\frac{2}{6}+\frac{6}{12}\right)$$
$$=\frac{1}{7}\left(\frac{9 + 4 + 6}{12}\right)$$
$$=\frac{1}{7}\left(\frac{19}{12}\right)=\frac{19}{84}$$

2. Given that a 7-digit number $$\overline{13xy45z}$$ is divisible by $$792$$. Find $$𝑦$$

$$792 = 8 × 9 × 11$$
Karena $$\overline{13xy45z}$$ habis dibagi $$792$$ maka $$\overline{13xy45z}$$ habis dibagi $$8, 9$$ dan juga $$11$$

• Syarat habis dibagi $$8$$
3 angka terakhir habis dibagi $$8$$, $$\overline{45z}$$ habis dibagi $$8$$, nilai $$z$$ yang memenuhi adalah $$6$$
• Syarat habis dibagi $$9$$
Jumlah angkanya habis dibagi $$9$$, maka $$1 + 3 + 𝑥 + 𝑦 + 4 + 5 + 6 = 19 + 𝑥 + 𝑦 = 27$$,
diperoleh nilai $$𝑥 + 𝑦 = 8$$, pasangan $$(x,y)$$ yang memenuhi adalah $$(0,8), (1,7), (2,6), (3,5), (4,4), (5,3), (6,2), (7,1), (8,0)$$
• Syarat habis dibagi $$11$$,
Selisih jumlah bilangan diposisi ganjil dan genap bernilai kelipatan $$11$$

$$(1 + 𝑥 + 4 + 6) − (3 + 𝑦 + 5) = 𝑘. 11$$
$$(11 + 𝑥) − (8 + 𝑦) = 𝑘. 11$$
$$3 + 𝑥 − 𝑦 = 𝑘. 11$$

Kemungkinan nilai $$𝑥 − 𝑦 = \{−3, 8\}$$, yang memenuhi $$(x,y)=(8, 0)$$.

Jadi diperoleh nilai $$y$$ yang memenuhi adalah $$0$$

3. How many triangles are there in the picture

A) 70
(B) 74
(C) 75
(D) 76

Bagi ke dalam bentuk 3 model segitiga, sama sisi, sama kaki dan siku-siku

Jadi banyak segitiga ada $$16 + 24 + 36 = 76$$

4. A circle is divided into three identical sectors by three radii. What is the proportion of the perimeter of a sector to its circumference?

(A) $$\frac{1}{3}$$
(B) $$\frac{1}{3}+\frac{1}{π}$$
(C) $$\frac{1}{π}$$
(D) $$\frac{1}{3}+\frac{2}{π}$$

Keliling lingkaran : $$2𝜋𝑟 = 2𝜋(3) = 6𝜋$$
Keliling juring lingkaran : $$\frac{1}{3}(6𝜋) + 3 + 3 = 2𝜋 + 6$$

Jadi perbandingan keliling juring dan keliling lingkaran adalah

$$(2𝜋 + 6) ∶ 6𝜋$$
$$(𝜋 + 3) ∶ 3𝜋$$

atau dapat ditulis $$\frac{1}{3}+\frac{1}{π}$$

5. Observe the pattern of the numbers and find “♡”

(A) 6
(B) 7
(C) 8
(D) 9

45 = 25 + 20
25 = 17 + 8

dengan mengikuti pola penjumlahan di atas nilai “♡” adalah 8

6. If the positive integer $$n$$ satisfies $$5 <\frac{1}{𝑛}+\frac{2}{𝑛}+\frac{3}{𝑛}+ ⋯ +\frac{15}{𝑛}< 8$$, how many such $$“n”$$ are there?

(A) 12
(B) 10
(C) 8
(D) 5

Untuk

$$5 <\frac{1}{𝑛}+\frac{2}{𝑛}+\frac{3}{𝑛}+ ⋯ +\frac{15}{𝑛}$$
$$5 <\frac{120}{𝑛}⇒ 𝑛 <\frac{120}{5}⇒ 𝑛 < 24$$

Untuk

$$\frac{1}{𝑛}+\frac{2}{𝑛}+\frac{3}{𝑛}+ ⋯ +\frac{15}{𝑛}< 8$$
$$\frac{120}{𝑛}< 8 ⇒\frac{120}{8}< 𝑛 ⇒ 𝑛 > 15$$

Diperoleh batasan nilai $$n$$ adalah

$$15 < 𝑛 < 24$$

Nilai $$𝑛$$ yang memenuhi adalah $$\{16, 17, 18, 19, 20, 21, 22, 23\}$$, jadi nilai $$𝑛$$ yang memenuhi banyaknya ada $$8$$

7) A cargo ship delivers goods in the speed of 28km/hr between two harbors A and B which are located at the upper reaches and the lower reaches of a river. If the downstream trip and the upstream trip last 5 hr and 9 hr, respectively, find the distance between the two harbors in km.

(A) 196
(B) 180
(C) 144
(D) 135

Misalkan kecepatan arus sungai adalah $$x$$

𝐽𝑎𝑟𝑎𝑘 𝐴 𝑘𝑒 𝐵 = 𝐽𝑎𝑟𝑎𝑘 𝐵 𝑘𝑒 𝐴
$$(28 + 𝑥)5 = (28 − 𝑥)9$$
$$140 + 5𝑥 = 252 − 9𝑥$$
$$14𝑥 = 112$$
$$𝑥 = 8$$

Jadi jarak $$A$$ ke $$B$$ adalah $$(28 − 𝑥)9 = (28 − 8)9 = 20(9) = 180$$ 𝑘𝑚

8) Given a regular sequence $$3, 4, 10, 24, 49, 88, x, ….$$ What should $$x$$ be?

(A) 100
(B) 107
(C) 121
(D) 144

$$3, 4, 10, 24, 49, 88, x$$

Pola beda tingkat pertama : $$1, 6, 14, 25, 39, a$$
Pola beda tingkat kedua : $$5, 8, 11, 14, 17$$
Diperoleh nilai $$𝑎 = 39 + 17 = 56$$

Dengan mengikuti pola di atas, nilai $$x$$ adalah $$88 + 56 = 144$$

9) A 3-digit number happens to be the cube of half of the sum of its three digits. Find the tens digit of this 3-digit number

(A) 1
(B) 2
(C) 4
(D) 5

10. As in the picture, fill 0～9 in the number sentences below. Each contains only a 1-digit number, and a 2-digit number should be filled in a linked or . Find A＋B＋C＋D.

(A) 29
(B) 27
(C) 24
(D) 20