OSN Sekolah Menengah Pertama

contoh soal

1. Berapa hasil dari $\sqrt{200\cdot 201\cdot 202\cdot 203)+1}$?

Jawaban

$$\begin{array}{cc}\sqrt{200\cdot 201\cdot 202\cdot 203)+1}& =\sqrt{200(200+1)(200+2)(200+3)+1}\\ \text{misalkan 200=x }\\ & =\sqrt{x(x+1)(x+2)(x+3)+1}\\ & =\sqrt{x(x+3)(x+1)(x+2)+1}\\ & =\sqrt{(x^2+3x)(x^2+3x+2)+1}\\ \text{misalkan }{x}^2+3x=n\\ & =\sqrt{n(n+2)+1}\\ & =\sqrt{n^2+2n+1}\\ & =\sqrt{(n+1{)}^2}\\ & =n+1\\ & =x^2+3x+1\\ & =200^2+3(200)+1\\ & =40.000+600+1\\ & =40.601\end{array}$$

2. Berapa hasil dari $\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\dots +\frac{1}{2023\times 2024}$?

Jawaban

$$\begin{array}{c}\text{cara 1 }\\ \text{Perhatian }\frac{1}{n\times (n+1)}& =\frac{1}{n}-\frac{1}{n+1}\\ \frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\dots +\frac{1}{2023\times 2024}& =(\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+\dots +(\frac{1}{2023}-\frac{1}{2024})\\ & =1-\frac{1}{2024}\\ & =\frac{2024}{2024}-\frac{1}{2024}\\ & =\frac{2024-1}{2024}\\ & =\frac{2023}{2024}\\ \text{cara 2 }\\ \frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\dots +\frac{1}{n\times (n+1)}& =\frac{n}{n+1}\\ \frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\dots +\frac{1}{2023\times 2024}& =\frac{2023}{2024}\end{array}$$

3. Berapa hasil dari $\frac{1}{5\times 8}+\frac{1}{8\times 11}+\frac{1}{11\times 14}+\dots +\frac{1}{62\times 65}$?

Jawaban

$$\begin{array}{cc}\text{misalkan }S& =\frac{1}{5\times 8}+\frac{1}{8\times 11}+\frac{1}{11\times 14}+\dots +\frac{1}{62\times 65}\\ S& =\frac{1}{5\times 8}+\frac{1}{8\times 11}+\frac{1}{11\times 14}+\dots +\frac{1}{62\times 65}\\ 3S& =\frac{3}{5\times 8}+\frac{3}{8\times 11}+\frac{3}{11\times 14}+\dots +\frac{3}{62\times 65}\\ & =\frac{1}{5}-\frac{1}{8}+(\frac{1}{8}-\frac{1}{11})+(\frac{1}{11}-\frac{1}{14})+\dots +(\frac{1}{62}-\frac{1}{65})\\ & =\frac{1}{5}-\frac{1}{65}\\ & =\frac{12}{65}\\ S& =\frac{1}{3}\times \frac{12}{65}\\ & =\frac{4}{65}\end{array}$$

4. Berapa hasil dari $\sqrt{6+\sqrt{6+\sqrt{6+\dots }}}$?

Jawaban

$$\begin{array}{cc}\sqrt{6+\sqrt{6+\sqrt{6+\dots }}}& =x\\ (\sqrt{6+\sqrt{6+\sqrt{6+\dots }}}{)}^2& =x^2\\ 6+(\sqrt{6+\sqrt{6+\dots }})& =x^2\\ 6+x& =x^2\\ x^2-x-6& =0\\ (x-3)(x-2)& =0\\ x=3& \text{ atau }x=-2\\ \text{ jadi x adalah }3\end{array}$$

5. Berapa hasil dari $\sqrt{20-\sqrt{20-\sqrt{20-\dots }}}$?

Jawaban

$$\begin{array}{cc}\sqrt{20-\sqrt{20+\sqrt{20-\dots }}}& =x\\ (\sqrt{20-\sqrt{20-\sqrt{20-\dots }}}{)}^2& =x^2\\ 20-(\sqrt{20-\sqrt{20-\dots }})& =x^2\\ 20-x& =x^2\\ x^2+x-20& =0\\ (x-4)(x+5)& =0\\ x=4& \text{ atau }x=-5\\ \text{ jadi x adalah }4\end{array}$$

6. Berapa hasil dari $\sqrt{2\sqrt{2\sqrt{2\dots }}}$?

Jawaban

$$\begin{array}{cc}\sqrt{2\sqrt{2\sqrt{2\dots }}}& =x\\ 2\sqrt{2\sqrt{2\dots }}& =x^2\\ \text{maka menjadi }\frac{x^2}{x}& =\frac{2\sqrt{2\sqrt{2\dots }}}{\sqrt{2\sqrt{2\sqrt{2\dots }}}}\\ x& =2\end{array}$$

7. Berapa hasil dari $\sqrt{\frac{8}{\sqrt{\frac{8}{\sqrt{\frac{8}{\dots }}}}}}$?

Jawaban

$$\begin{array}{cc}\sqrt{\frac{8}{\sqrt{\frac{8}{\sqrt{\frac{8}{\dots }}}}}}& =x\\ \frac{8}{\sqrt{\frac{8}{\sqrt{\frac{8}{\dots }}}}}& =x^2\\ \frac{8}{x}& =x^2\\ x^3& =8\\ x& =\sqrt[3]{8}\\ x& =2\end{array}$$

8. Berapa hasil dari $\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}$?

Jawaban

$$\begin{array}{c}\text{cara 1}\\ \sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}& =\sqrt{3\sqrt{3\sqrt{3\times 3^{\frac{1}{2}}}}}\\ & =\sqrt{3\sqrt{3\sqrt{3^{\frac{3}{2}}}}}\\ & =\sqrt{3\sqrt{3\times 3^{\frac{3}{4}}}}\\ & =\sqrt{3\sqrt{3^{\frac{7}{4}}}}\\ & =\sqrt{3\times 3^{\frac{7}{8}}}\\ & =\sqrt{3^{\frac{15}{8}}}\\ & =3^{\frac{15}{16}}\\ & =\sqrt[16]{3^{15}}\\ \text{cara 2}\\ \text{Gunakan rumus }{a}^{\frac{2^n-1}{2^n}}\text{ n adalah banyaknya akar }\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}& =3^{\frac{2^4-1}{2^4}}\\ & =3^{\frac{16-1}{16}}\\ & =3^{\frac{15}{16}}\\ & =\sqrt[16]{3^{15}}\end{array}$$

9. Berapa nilai x dari $\sqrt{4x+\sqrt{4x+\sqrt{4x+\dots }}}=9$?

Jawaban

$$\begin{array}{cc}\sqrt{4x+\sqrt{4x+\sqrt{4x+\dots }}}& =9\\ (\sqrt{4x+\sqrt{4x+\sqrt{4x+\dots }}}{)}^2& =(9{)}^2\\ 4x+(\sqrt{4x+\sqrt{4x+\dots }})& =81\\ 4x+9& =81\\ 4x& =72\\ x& =13\end{array}$$

10. Berapa nilai x dari $\sqrt{7x+2-\sqrt{7x+2-\sqrt{7x+2-\dots }}}=12$?

Jawaban

$$\begin{array}{cc}\sqrt{7x+2-\sqrt{7x+2-\sqrt{7x+2-\dots }}}& =12\\ (\sqrt{7x+2-\sqrt{7x+2-\sqrt{7x+2-\dots }}}{)}^2& =(12{)}^2\\ 7x+2-(\sqrt{7x+2-\sqrt{7x+2-\dots }})& =144\\ 7x+2-12& =144\\ 7x& =154\\ x& =22\end{array}$$

11. Berapa nilai x dari $\sqrt{72-\sqrt{72+\sqrt{72+\dots +\sqrt{x}}}}=9$?

Jawaban

$$\begin{array}{cc}\sqrt{72-\sqrt{72-\sqrt{72-\dots -\sqrt{x}}}}& =9\\ (\sqrt{72-\sqrt{72-\sqrt{72-\dots -\sqrt{x}}}}{)}^2& =9^2\\ 72-\sqrt{72-\sqrt{72-\dots -\sqrt{x}}}& =81\\ \sqrt{72-\sqrt{72-\dots -\sqrt{x}}}& =9\\ \text{ karena terus berulang maka }\\ \sqrt{x}& =9\\ x& =81\end{array}$$

12. Berapa nilai x dari $\sqrt{121-\sqrt{121-\sqrt{121-\dots -\sqrt{x}}}}=10$?

Jawaban

$$\begin{array}{cc}\sqrt{121-\sqrt{121-\sqrt{121-\dots -\sqrt{x}}}}& =10\\ (\sqrt{12--\sqrt{121-\sqrt{121-\dots -\sqrt{x}}}}{)}^2& ={10}^2\\ 121-\sqrt{121-\sqrt{121-\dots -\sqrt{x}}}& =100\\ \sqrt{121-\sqrt{121-\dots -\sqrt{x}}}& =21\\ \text{ karena terus berulang maka }\\ \sqrt{x}& =21\\ x& =441\end{array}$$

13. Berapa hasil dari $\frac{1}{2+3\frac{1}{2+3\frac{1}{2+3\dots }}}$?

Jawaban

$$\begin{array}{c}\frac{1}{2+3\frac{1}{2+3\frac{1}{2+3\dots }}}=\\ \text{Misalkan }\frac{1}{2+3\frac{1}{2+3\dots }}& =x\\ \frac{1}{2+3x}& =x\\ 1& =x(2+3x)\\ 1& =2x+3x^2\\ 3x^2+2x-1& =0\\ (3x-1)(x+1)& =0\\ x=\frac{1}{3}& \text{ atau }x=-1\end{array}$$

14. Berapa hasil dari $7+\frac{16}{1+\frac{56}{1+\frac{56}{1+\frac{56}{1+\dots }}}}$?

Jawaban

$$\begin{array}{c}7+\frac{16}{1+\frac{56}{1+\frac{56}{1+\frac{56}{1+\dots }}}}=\\ \text{Misalkan }1+\frac{56}{1+\dots }& =x\\ 1+\frac{56}{x}& =x\\ x+56& =x^2\\ x^2-x-56& =0\\ (x-8)(x+7)& =0\\ x=8& \text{ atau }x=-7\\ \text{Karena hasilnya selalu bilangan positif jadi }x=8\\ 7+\frac{16}{8}& =7+2=9\end{array}$$

15. Berapa hasil dari $x^2-3xy+y^2$ jika x+y = 7 dan xy = -4?

Jawaban

$$\begin{array}{cc}{x}^2-3xy+y^2& =x^2+2xy+y^2-5xy\\ & =(x+y{)}^2-5xy\\ & =7^2-5(-4)\\ & =49+20\\ & =69\end{array}$$

16. Berapa hasil dari $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}$ jika x+y+z = 2961 dan $\frac{1}{x+y}+\frac{1}{x+z}+\frac{1}{y+z}=\frac{1}{7}$?

Jawaban

$$\begin{array}{cc}\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}& =\frac{x}{y+z}+1+\frac{y}{x+z}+1+\frac{z}{x+y}+1-3\\ & =\frac{x+y+z}{y+z}+\frac{x+y+z}{x+z}+\frac{x+y+z}{x+y}-3\\ & =(x+y+z)(\frac{1}{y+z}+\frac{1}{x+z}+\frac{1}{x+y})-3\\ & =2961(\frac{1}{7})-3\\ & =423-3\\ & =420\end{array}$$

17. Berapa hasil dari $\frac{2027\times (2025^2-9)\times 2023}{2028\times (2025^2-4)}$?

Jawaban

$$\begin{array}{c}\frac{2027\times (2025^2-9)\times 2023}{2028\times (2025^2-4)}\\ \text{misalkan x=2025 }\\ \frac{(x+2)\times (x^2-9)\times (x-2)}{(x+3)\times (x^2-4)}\\ \frac{(x-3)\times (x+3)\times (x^2-4)}{(x+3)\times (x^2-4)}\\ x-3\\ 2025-3\\ 2022\end{array}$$

18. Berapa hasil x dari $\frac{x-10}{2023}+\frac{x-9}{2024}+\frac{x-8}{2025}=3$?

Jawaban

$$\begin{array}{cc}\frac{x-10}{2023}+\frac{x-9}{2024}+\frac{x-8}{2025}& =3\\ \frac{x-10}{2023}+\frac{x-9}{2024}+\frac{x-8}{2025}& =1+1+1\\ \frac{x-10}{2023}-1+\frac{x-9}{2024}-1+\frac{x-8}{2025}-1& =0\\ \frac{x-10-2023}{2023}+\frac{x-9-2024}{2024}+\frac{x-8-2025}{2025}& =0\\ \frac{x-2033}{2023}+\frac{x-2033}{2024}+\frac{x-2033}{2025}& =0\\ (x-2033)(\frac{1}{2023}+\frac{1}{2024}+\frac{1}{2025})& =0\\ x-2033& =0\\ x& =2033\end{array}$$

19. Berapa hasil x dari $\frac{x-17}{2026}+\frac{x-19}{2024}+\frac{x-21}{2022}=3$?

Jawaban

$$\begin{array}{cc}\frac{x-17}{2026}+\frac{x-19}{2024}+\frac{x-21}{2022}& =3\\ \frac{x-17}{2026}+\frac{x-19}{2024}+\frac{x-21}{2022}& =1+1+1\\ \frac{x-17}{2026}-1+\frac{x-19}{2024}-1+\frac{x-21}{2022}-1& =0\\ \frac{x-17-2026}{2026}+\frac{x-19-2024}{2024}+\frac{x-21-2022}{2022}& =0\\ \frac{x-2043}{2026}+\frac{x-2043}{2024}+\frac{x-2043}{2022}& =0\\ (x-2043)(\frac{1}{2026}+\frac{1}{2024}+\frac{1}{2022})& =0\\ x-2043& =0\\ x& =2043\end{array}$$

20. Berapa hasil x dari $\frac{x-4}{674}+\frac{x-4}{1011}+\frac{x-1}{2025}=6$?

Jawaban

$$\begin{array}{cc}\frac{x-4}{674}+\frac{x-4}{1011}+\frac{x-1}{2025}& =6\\ \frac{x-4}{674}+\frac{x-4}{1011}+\frac{x-1}{2025}& =3+2+1\\ \frac{x-4}{674}-3+\frac{x-4}{1011}-2+\frac{x-1}{2025}-1& =0\\ \frac{x-4-2022}{674}+\frac{x-4-2022}{1011}+\frac{x-1-2025}{2025}& =0\\ \frac{x-2026}{674}+\frac{x-2026}{1011}+\frac{x-2026}{2025}& =0\\ (x-2026)(\frac{1}{674}+\frac{1}{1011}+\frac{1}{2025})& =0\\ x-2026& =0\\ x& =2026\end{array}$$

21. Berapa hasil x dari $\frac{x-1}{2024}+\frac{x-2}{2023}+\frac{x-3}{2022}+\frac{x-2040}{5}=0$?

Jawaban

$$\begin{array}{cc}\frac{x-1}{2024}+\frac{x-2}{2023}+\frac{x-3}{2022}+\frac{x-2040}{5}& =0\\ \frac{x-1}{2024}-1+\frac{x-2}{2023}-1+\frac{x-3}{2022}-1+\frac{x-2040}{5}+3& =0\\ \frac{x-1-2024}{2024}+\frac{x-2-2023}{2023}+\frac{x-3-2022}{2022}+\frac{x-2040+15}{5}& =0\\ \frac{x-2025}{2024}+\frac{x-2025}{2023}+\frac{x-2025}{2022}+\frac{x-2025}{5}& =0\\ (x-2025)(\frac{1}{2024}+\frac{1}{2023}+\frac{1}{2022}+\frac{1}{5})& =0\\ x-2025& =0\\ x& =2025\end{array}$$

22. Berapa hasil x dari $\frac{11-x}{2029}+\frac{10-x}{2030}+\frac{9-x}{2031}+\frac{2070-x}{10}=0$?

Jawaban

$$\begin{array}{cc}\frac{11-x}{2029}+\frac{10-x}{2030}+\frac{9-x}{2031}+\frac{2070-x}{10}& =0\\ \frac{11-x}{2029}+1+\frac{10-x}{2030}+1+\frac{9-x}{2031}+1+\frac{2070-x}{10}-3=0\\ \frac{11-x+2029}{2029}+\frac{10-x+2030}{2030}+\frac{9-x+2031}{2031}+\frac{2070-x-30}{10}=0\\ \frac{2040-x}{2029}+\frac{2040-x}{2030}+\frac{2040-x}{2031}+\frac{2040-x}{10}=0\\ (2040-x)(\frac{1}{2029}+\frac{1}{2030}+\frac{1}{2031}+\frac{1}{10})& =0\\ 2040-x& =0\\ x& =2040\end{array}$$

23. Berapa nilai maksimum dari fungsi f(x,y) = 3x+4y jika x²+y²=100 serta x dan y adalah bilangan bulat positif?

Jawaban

$$\begin{array}{c};\text{cara 1}\\ \text{ ada beberapa kemungkinan jika }{x}^2+y^2=100\text{ yaitu }(0,10),(6,8)\\ \text{coba-coba untuk mengetahui hasilnya }\\ 3(0)+4(10)& =40\\ 3(10)+4(0)& =30\\ 3(6)+4(8)& =50\\ 3(8)+4(6)& =48\\ \text{ jadi nilai maksimum adalah 50 }\\ ;\text{cara 2}\\ x^2+y^2& =100\\ (10cosa{)}^2+(10sina{)}^2& =100\\ 3x+4y& =3(10cosa)+4(10sina)\\ & =30cosa+40sina\\ & =50(\frac{3}{5}cosa+\frac{4}{5}sina)\\ & =50(sinb\cdot cosa+cosb\cdot sina)\\ & =50sin(b+a)\\ \text{ jadi nilai maksimum adalah 50 }\\ \text{cara 3}\\ \text{rumus}\\ (ax+by{)}^2\le (a^2+b^2)(x^2+y^2)\\ -\sqrt{(a^2+b^2)(x^2+y^2)}\le ax+by\le \sqrt{(a^2+b^2)(x^2+y^2)}\\ -(3^2+4^2)(100)\le (3x+4y{)}^2\le (3^2+4^2)(100)\\ -(25)(100)\le (3x+4y{)}^2\le (25)(100)\\ -2.500\le (3x+4y{)}^2\le 2.500\\ -\sqrt{2.500}\le 3x+4y\le 50\\ \text{ jadi nilai maksimum adalah 50 }\end{array}$$

24. Berapa nilai minimum x+y+z jika xyz=64?

Jawaban

$$\begin{array}{cc}xyz& =64\\ \sqrt[3]{xyz}& =\sqrt[3]{64}\\ \sqrt[3]{xyz}& =4\\ \text{ dengan mengingat rumus rataan geometri dan rataan aritmatika }\\ \sqrt[3]{xyz}& =\frac{x+y+z}{3}\\ 4& =\frac{x+y+z}{3}\\ x+y+z& =12\end{array}$$

25. Berapa banyaknya bilangan kurang dari atau sama dengan 50 yang memiliki 6 faktor?

menggunakan pola bilangan prima seperti mencari kpk dan fpb.

kemungkinan pertama: p⁵ maka hanya 2⁵ = 32 saja

kemungkinan kedua: p²q maka beberapa kemungkinan sebagai berikut:

2²3 = 12, 2²5 = 20, 2²7 = 28, 2²11 = 44

3²2 = 18, 3²5 = 45

5²2 = 50

jadi banyaknya adalah 8.

26. Dua dadu dilempar bersama-sama satu kali. Berapa peluang bahwa dua dadu yang muncul berangka sama?

• jumlah seluruh dadu (s) yaitu 6x6 = 36
• dua dadu yang sama angkanya (a) yakni {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} jadi ada 6
• maka peluangnya adalah $P(a)=\frac{6}{36}=\frac{1}{6}$

27. Jumlah kedua bilangan adalah 30 maka berapa nilai maksimum dari hasil kali kedua bilangan?>/li>

• Jumlah kedua bilangan yang menghasilkan 30 yang mungkin adalah (0,30), (1,29), (2,28), (3,27), …., (15,15)
• Hasil kali kedua bilangan masing-masing yakni 0, 29, 56, 81, 104, 125, …., 225
• Jadi hasil kali yang paling maksimum adalah 225

28. Berapa angka desimal ke 2024 jika hasil dari 1/7?

• Hasil dari 1/7 adalah 0,142857142857…
• Karena berulang-ulang keenam angka yang sama maka sisa dari 2024 dibagi 6 yaitu 0. angka 0 berarti 7.

29. Berapa angka desimal ke 2024 jika hasil dari 1/13?

• Hasil dari 1/13 adalah 0,076923076923…
• Karena berulang-ulang keenam angka yang sama maka sisa dari 2024 dibagi 6 yaitu 0. angka 0 berarti 3.

30. Dua persamaan yaitu 43a+20b-10c=36 dan 2a-2b+19c=-9 maka berapa hasil dari 5a+2b+c?

Jawaban

$$\begin{array}{cc}43a+20b-10c& =36\\ 2a-2b+19c& =-9\\ 45a+18b+9c& =27\text{ (persamaan (1) ditambahkan (2))}\\ 5a+2b+c& =3\end{array}$$

31. Berapa hasil f(16)-f(7) dari f(3x-2)=4x-7?

Jawaban

$$\begin{array}{cc}f(16)& =f(3x-2)\\ 16& =3x-2\\ 3x& =18\\ x& =6\\ f(16)& =4(6)-7\\ & =17\\ f(7)& =f(3x-2)\\ 7& =3x-2\\ 3x& =9\\ x& =3\\ f(7)& =4(3)-7\\ & =5\\ f(16)-f(7)& =17-5\\ & =12\end{array}$$

32. Sebuah persegi memiliki dua persegi panjang secara sembarangan baik vertikal atau horisontal. jika keliling kedua persegi panjang adalah 102 meter maka berapa luas persegi?

Jawaban

$$\begin{array}{cc}b& =a+c\\ k& =2(a+b)+2(b+c)\\ 102& =2a+4b+2c\\ 51& =a+2b+c\\ 51& =b+2b\\ 51& =3b\\ b& =17\\ l& =b^2\\ & ={17}^2\\ & =289m^2\end{array}$$