making this blog great again, 111 problem at a time

by wu2481632, Jan 13, 2017, 6:26 PM

Prove that there is no integer $n \ge 2$ for which
$$\dfrac{3^n - 2^n}{n}$$is an integer. (111)

Solution

Determine all positive integers relatively prime to all the terms of the infinite sequence
$$a_n = 2^n + 3^n + 6^n - 1, n\ge 1.$$(111, 2005 IMO P4)

Solution

A small piece of tasty food
This post has been edited 1 time. Last edited by shiningsunnyday, Jan 14, 2017, 2:59 AM
Reason: Typo fixed and source of second problem added

Comment

7 Comments

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For Case 2 on the first question, you should replace $n$ with an arbitrary prime $p$ which is a divisor of $n$.

by WalkerTesla, Jan 13, 2017, 6:51 PM

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Alternate solution for the first question

by pi_Plus_45x23, Jan 13, 2017, 7:20 PM

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Oh yeah you're right @walker

SSD pls change? Thanks

by wu2481632, Jan 13, 2017, 8:32 PM

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Hey wu how hard does 111 get?

by shiningsunnyday, Jan 14, 2017, 3:00 AM

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111 is relatively easy; I think it would be too easy for you by far. The second problem I posted is something around 40~ish on the NT section out of 56...this book at least might help me learn ineq i guess...

by wu2481632, Jan 14, 2017, 4:30 AM

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w8 don't u h8 ineq with a passion

by zephyrcrush78, Jan 14, 2017, 4:35 AM

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hi, the solution to #1 is false?

in particular, $2^n \equiv 2\pmod p$ is not necessarily true (take $p = 5, n = 35$, for example, then $2^n = 2^3 \pmod p$)

however, you can note that asdf

by MathStudent2002, Jan 18, 2017, 12:50 AM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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