MathDB

2021 Auckland Mathematical Olympiad

Part of Auckland Mathematical Olympiad

Subcontests

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looking for the 4-digit password (2021 Auckland MO J3)

Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: ∙\bullet it is a 44-digit number whose first digit is 55. ∙\bullet it is a multiple of 99; ∙\bullet The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}} = rational (2021 Auckland MO S3)

For how many integers nn between 1 1 and 20212021 does the infinite nested expression n+n+n+...\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}} give a rational number?
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