Olympiad Maths Prep

Track / Stage 5 / 203 of 400 #803 of 2000

Problem 803

AIME late
Number theory Difficulty 5.5 Find the answer

2. 求以下二次无理数的循环连分数表示式, 它的纯循环部分及周期: □\square
(i) (5+37)/3(5+\sqrt{37}) / 3;
(ii) 43\sqrt{43};
(iii) (6+43)/7(6+\sqrt{43}) / 7;
(iv) 80+8\sqrt{80}+8;
(v) (3+7)/2(3+\sqrt{7}) / 2;
(vi) 26/5\sqrt{26 / 5}.

Official solution

2. (i) ⟨3,1,2‾⟩\langle\overline{3,1,2}\rangle;
(ii) ⟨6,1,1,3,1,5,1,3,1,1,12‾⟩\langle 6, \overline{1,1,3,1,5,1,3,1,1,12}\rangle;
(iii) ⟨1,1,3,1,5,1,3,1,1,43+6⟩=⟨1,1,3,1,5,1,3,1,1,12‾⟩\langle 1,1,3,1,5,1,3,1,1, \sqrt{43}+6\rangle=\langle\overline{1,1,3,1,5,1,3,1,1,12}\rangle;
(iv) ⟨16,1‾⟩\langle\overline{16,1}\rangle; (v) ⟨2,1,4,1,1‾⟩\langle 2, \overline{1,4,1,1}\rangle; (vi) ⟨2,3,1,1,3,4‾⟩\langle 2, \overline{3,1,1,3,4}\rangle.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.