# WMI Preliminary Round 2022 [Grade 8A]

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 8A tahun 2022

1) Given that $$40^2=1600$$ and $$50^2=2500$$. If $$n$$ is an integer, and $$n<\sqrt{2022}<n+1$$, find $$n$$.

(A) 47
(B) 45
(C) 44
(D) 43

$$π < \sqrt{2022} < π + 1$$
$$\sqrt{44^2} < \sqrt{2022} < \sqrt{2025}$$
$$\sqrt{44^2} < \sqrt{2022} < \sqrt{45^2}$$
$$44 < \sqrt{2022} < 45$$

Sehingga diperoleh nilai $$n$$ yang memenuhi adalah $$44$$

2) Find the hundreds digit of $$9999.98Γ9999.98$$

(A) 4
(B) 5
(C) 6
(D) 8

3) Which option below is the factor of $$2(x-1)^2+5(x-1)+3$$?

(A) x
(B) x-1
(C) x+1
(D) 2x+3

4) In the picture a rectangle ABCD is $$3x+1$$ in lenght and $$2x+1$$ in width. If a square whose side length is 3 is cut from the rectangle, find the perimeter of the remaining part.

(A) 10x
(B) 10x-2
(C) 10x-4
(D) 10x+4

Keliling bangun yang terpotong sama saja dengan keliling persegi panjang ABCD

$$πΎπππππππ = 2(π + π) = 2(3π₯ + 1 + 2π₯ + 1) = 2(5π₯ + 2) = 10π₯ + 4$$

5) A pair of set squares are placed as below. If $$β I=80ΒΊ$$, find $$β 2=?$$

(A) 80ΒΊ
(B) 95ΒΊ
(C) 100ΒΊ
(D) 105ΒΊ

Dengan menggunakan sifat garis lurus, sudut saling bertolak belakang dan jumlah sudut segitiga diperoleh $$β 2 = 95Β°$$

6) In $$Ξπ΄π΅πΆ, AB= \sqrt{6} + 1, BC = 2 + \sqrt{3}$$, and $$AC= \sqrt{2} + \sqrt{5}$$, Find the relation among $$β π΄,β π΅,$$ and $$β πΆ$$.

(A) $$β A>β B>β C$$
(B) $$β B>β A>β C$$
(C) $$β C>β B>β A$$
(D) $$β A>β C>β B$$

$$π΄B= \sqrt{6} + 1 β π΄π΅^2 = 6 + 1 + 2\sqrt{6} = 7 + \sqrt{24}$$
$$π΅C= 2 + \sqrt{3} β π΅πΆ^2 = 4 + 3 + 4\sqrt{3} = 7 + \sqrt{48}$$
$$π΄C= \sqrt{2} + \sqrt{5} β π΄πΆ^2 = 2 + 5 + 2\sqrt{10} = 7 + \sqrt{40}$$
Hubungan panjang sisi $$π΅C > π΄C > AB$$,karena panjang sisi berbanding lurus dengan besar sudut maka hubungan ketiga sudut segitiga adalah $$β π΄ > β π΅ > β πΆ$$

7) Supose $$π, π, 48, π, π$$ and $$30$$ make an arithmetic sequence; $$π, π₯, π,$$ and $$π¦$$ make a geometric
sequence. Find $$π¦$$.

(A) $$12\sqrt{3}$$
(B) $$12\sqrt{6}$$
(C) $$18\sqrt{2}$$
(D) $$18\sqrt{6}$$

$$π, π, 48, π, π$$ and $$30$$ adalah barisan aritmatika dengan beda = $$π$$
$$π = 48 β 2π$$
$$π = 48 β π$$
$$π = 48 + π$$
$$π = 48 + 2π$$
$$30 = 48 + 3π β π = β6$$
$$π, π₯, π,$$ and $$π¦$$ membentuk barisan geometri, maka berlaku
$$\frac{π₯}{π}=\frac{π}{π₯}=\frac{π¦}{π}$$
$$\frac{π₯}{48 β π}=\frac{48 + 2π}{π₯}$$
$$β\frac{π₯}{54}=\frac{36}{π₯}$$
$$β π₯^2 = 54(36) β π₯ = \sqrt{9.6.36} = 18\sqrt{6}$$
Selanjutnya
$$\frac{π}{π₯}=\frac{π¦}{π}$$
$$β\frac{36}{18\sqrt{6}}=\frac{π¦}{36}$$
$$β π¦=\frac{36(36)}{18\sqrt{6}}=\frac{72}{\sqrt{6}}Γ\frac{\sqrt{6}}{\sqrt{6}}=12\sqrt{6}$$

8) As shown in the picture, $$πΏ//π, β 1 = β 3 = 35Β°$$. Find $$β 1 + β 2 + β 3 + β 4 + β 5$$.

(A) 180ΒΊ
(B) 200ΒΊ
(C) 210ΒΊ
(D) 240ΒΊ

Pertama tarik 3 garis lurus pada ketika titik sudut diantara garis L dan MΒ  yang sejajar garis L dan M, lalu gunakan sifat sudut sehadap sampai mendapatkan posisi β 1 + β 2 + β 3 dan bagian akhir gunakan sifat sudut bersebrangan dalam.

Dari gambar terbawah besar sudut $$β 1 + β 2 + β 3 + β 4 + β 5 = 180Β°$$

9) Two grid points A and B are on a piece of $$4Γ4$$ square paper. Find a grid point C to make $$ΞABC$$ an isosceles right triangle. How many such points C’s are there?

(A) 1
(B) 2
(C) 3
(D) 4

Jadi ada 3 titik sedemikian sehingga dapat membentuk segitiga siku-siku sama kaki

10) Company W uses the linear function to adjust employees’ salary. Below shows the monthly salary of three employees in 2021 and 2022. How much is Max’ monthly salary in 2021?

(A) $700 (B)$750
(C) $800 (D)$900

11. In the picture, $$ABCD$$ is a square, and $$AP = 7, BP = 13$$. Find the area of $$ABCD$$

(A) 240
(B) 256
(C) 288
(D) 289

misalkan $$π΄π = π₯$$, diperoleh $$ππ = π₯ β 7$$, gunakan rumus Pythagoras pada segitiga $$πππ΅$$

$$π΅π = \sqrt{ππ΅^2 + ππ^2} = \sqrt{π₯^2 + (π₯ β 7)^2} = 13$$
$$βπ₯^2 + π₯^2 β 14π₯ + 49 = 132$$
$$β2π₯^2 β 14π₯ + 49 β 169 = 0$$
$$β2π₯^2 β 14π₯ β 120 = 0$$
$$βπ₯^2 β 7π₯ β 60 = 0$$
$$β(π₯ β 12)(π₯ + 5) = 0$$
$$βπ₯ = 12$$

Selanjutnya Pythagoras $$AOB$$

$$π΄π^2 + π΅π^2 = π΄π΅^2$$
$$144 + 144 = π΄π΅^2 = 288$$

Jadi luas persegi adalah $$π΄π΅^2 = 288$$ satuan luas.

12. Set $$f(n)οΌ4nοΌ90$$, in which n is a positive integer. If $$f(1)οΌf(2)οΌf(3)οΌβ¦οΌf(n)οΌ0$$, find $$n$$.

(A) 34
(B) 36
(C) 44
(D) 46

$$f(1)οΌf(2)οΌf(3)οΌβ¦οΌf(n)οΌ0$$
$$β4(1) β 90 + 4(2) β 90 + 4(3) β 90 + β― + 4(π) β 90 = 0$$
$$β4(1 + 2 + 3 + β― + π) β 90π = 0$$
$$β4(\frac{π(π + 1)}{2}) = 90π$$
$$β2(π + 1) = 90$$
$$βπ + 1 = 45$$
$$βπ = 44$$

13) Set the median of the ten numbers 1, 3, 3, 4, 5, 5, 6, 7, 7, and 9 to be $$a$$. If a number is taken out at will from these ten numbers, find the probability that such number is larger than $$a$$.

(A) $$\frac{2}{5}$$
(B) $$\frac{3}{5}$$
(C) $$\frac{1}{4}$$
(D) $$\frac{2}{3}$$

14) Given that $$π$$ is a root of the quadratic equation $$π₯^2 + 2π₯ β 21 = 0$$. Find $$(π β 3)(π + 3)(π β1)(π + 5)$$

(A) 20
(B) -15
(C) -20
(D) -35

$$k$$ merupakan akar dari $$π₯^2 + 2π₯ β 21 = 0$$ maka berlaku $$π^2 + 2π β 21 = 0$$
selanjutnya

$$(π β 3)(π + 3)(π β 1)(π + 5) = (π β 3)(π + 5)(π β 1)(π + 3)$$
$$= (π^2 + 2π β 15)(π^2 + 2π β 3)$$
$$= (21 β 15)(21 β 3) = 6(18) = 108$$

15) Given a rhombus $$ABCD$$ on the rectangular coordinate plane. Suppose its side $$AD β₯ y-axis$$ on $$E$$, point $$B$$ is on $$y-axis, BC οΌ5, BE οΌ2 DE$$ , and the graph of the inverse function $$yοΌ\frac{k}{x}(xοΌ0)$$ passes through points $$C$$ and $$D$$ at the same time. Find $$k$$.

(A) $$\frac{20}{3}$$
(B) $$\frac{40}{3}$$
(C) $$\frac{5}{2}$$
(D) $$\frac{5}{4}$$

WMI Preliminary Round 2021 [Grade 10B]