OSN Sekolah Menengah Pertama

contoh soal

1. 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

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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

1. 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 $P(a)={\frac {6}{36}}={\frac {1}{6}}$

1. Jumlah kedua bilangan adalah 30 maka berapa nilai maksimum dari hasil kali kedua bilangan?

• 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

1. 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.

1. 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.

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

Jawaban

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

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

Jawaban

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