A removing number problem

11082012, 02:15 PM
(This post was last modified: 11082012 02:52 PM by elim.)
Post: #1




A removing number problem
source (Vietnam NMO 1990_2)
At least $ n  1$ numbers are removed from the set $ A = \{1, 2, \ldots, 2n  1\}$ according to the following rules: (i) If $ a$ is removed, so is $ 2a$; (ii) If $ a$ and $ b$ are removed, so is $ a + b$. Find the way of removing numbers such that the sum of the remaining numbers is maximum possible. 

11082012, 02:51 PM
Post: #2




RE: A removing number problem
We have to remove $n1$ even numbers $2,4,\ldots,2n2$ so the sum of the remaining is maximum!
The solution is from these Lemma: Lemma1: If we remove $1$ we have to remove $2,3,\ldots$ so we remove all numbers from the set $A$ Lemma2: If we remove $2$, we remove $n1$ numbers $2,4,\ldots,2n2$ so the sum of remaining numbers is $n^2$ Lemma3: If we remove $p$ number $a_1<a_2<a_3<\cdots <a_p$ so we must remove the number $a_1+a_p$ but this number greater than $a_p$ so $a_1+a_p > 2n1 $ A similar way to $a_i$ and $a_{p+1i}$. So the sum of the number which are removed is $\ge n(n1)$ So the sume of the remaining numbers is $< n^2$. 

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