IMO Shortlist 1991
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- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Find the largest power of $1991$ that divides$1990^{1991^{1992}}+1992^{1991^{1990}}$.
Re: IMO Shortlist 1991
Apply LTE twice: on \(1990^{1991^{1992}}+1\) and on \(1992^{1991^{1990}}-1\). The answer is \(1991\).
Last edited by Nirjhor on Fri Sep 19, 2014 11:03 pm, edited 1 time in total.
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: IMO Shortlist 1991
Nirjhor, elaborate please. And.. one more thing: $1991$ is not a prime.
Re: IMO Shortlist 1991
\(1991=11\times 181\). Work with each of these primes. Solution added to the previous comment.
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: IMO Shortlist 1991
$1990^{{1991}^{1992}}+1992^{{1991}^{1990}}=[{1990}^{{1991}^2}]^{{1991}^{1990}}+1992^{{1991}^{1990}}$
Now apply LTE twice.
Now apply LTE twice.
Re: IMO Shortlist 1991
No power of 1991 is divisible by 199019911992+199219911990.
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