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Difference between revisions of "2010 IMO Problems/Problem 5"

(Created page with '== Problem == Each of the six boxes <math>B_1</math>, <math>B_2</math>, <math>B_3</math>, <math>B_4</math>, <math>B_5</math>, <math>B_6</math> initially contains one coin. The f…')
 
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Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes <math>B_1</math>, <math>B_2</math>, <math>B_3</math>, <math>B_4</math>, <math>B_5</math> become empty, while box <math>B_6</math> contains exactly <math>2010^{2010^{2010}}</math> coins.
 
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes <math>B_1</math>, <math>B_2</math>, <math>B_3</math>, <math>B_4</math>, <math>B_5</math> become empty, while box <math>B_6</math> contains exactly <math>2010^{2010^{2010}}</math> coins.
  
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''Author: Unknown currently''
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''Author: Hans Zantema''
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== Solution ==
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Let the notation <math>[a_1,a_2,a_3,a_4,a_5,a_6]</math> be the configuration in which the <math>x</math>-th box has <math>a_x</math> coin,
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Let <math>T=2010^{2010^{2010}}</math>.
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<br/>
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Our starting configuration is <math>[1,1,1,1,1,1]</math>
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<br />
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We need these compound moves:
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Compound move 1: <math>[a,0]\rightarrow[0,2a]</math>, this is just repeated type 1 move on all <math>a</math> coins.
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Compound move 2: <math>[a,0,0]\rightarrow[0,2^a,0]</math>, apply type 1 move on 1 of the coin to get <math>[a-1,2,0]</math>, then apply compound move 1 to the 2 coins to get <math>[a-1,0,2^2]</math>, apply type 2 move and get <math>[a-2,2^2,0]</math>, and repeat compound move 1 and type 2 move until <math>[0,2^a,0]</math> is achieve.
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Compound move 3: <math>[a,b,0,0]\rightarrow[a-1,2^b,0,0]</math>, apply compound move 2 to obtain <math>[a,0,2^b,0]</math> and use type 2 move to get <math>[a-1,2^b,0,0]</math>
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Compound move 4: <math>[a,0,0,0]\rightarrow[0,2^{2^{.^{.^{.^2}}}},0,0]</math> with <math>a</math> <math>2</math>'s. Apply compound move 3 <math>a</math> times.
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Compound move 5: <math>[a,0,0]\rightarrow[b,0,0]</math> with <math>a>b</math>, use type 2 move <math>(a-b)</math> times.
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<br/><br/>
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Let's follow this move:
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<math>[1,1,1,1,1,1] \rightarrow[0,3,1,1,1,1] \rightarrow[0,2,3,1,1,1]\rightarrow[0,2,1,5,1,1]\rightarrow[0,2,1,1,9,1]\rightarrow[0,2,1,1,0,19]\rightarrow[0,1,19,0,0,0]</math>
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Using Compound move 1, 4 and 5, We can obtain:
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<math>[0,1,19,0,0,0]\rightarrow[0,1,0,X,0,0]\rightarrow[0,0,X,0,0,0]\rightarrow[0,0,0,Y,0,0]\rightarrow[0,0,0,T/4,0,0]\rightarrow[0,0,0,0,T/2,0]\rightarrow[0,0,0,0,0,T]</math>
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, where <math>X, Y = 2^{2^{.^{.^{.^2}}}}</math> where X has <math>19</math> 2's and Y has <math>X</math> 2's, and <math>Y</math> is clearly bigger then <math>T/4</math>
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== See Also ==
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{{IMO box|year=2010|num-b=4|num-a=6}}
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[[Category:Olympiad Geometry Problems]]

Latest revision as of 05:25, 25 June 2018

Problem

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;

Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.

Author: Hans Zantema

Solution

Let the notation $[a_1,a_2,a_3,a_4,a_5,a_6]$ be the configuration in which the $x$-th box has $a_x$ coin,

Let $T=2010^{2010^{2010}}$.


Our starting configuration is $[1,1,1,1,1,1]$ 


We need these compound moves:

Compound move 1: $[a,0]\rightarrow[0,2a]$, this is just repeated type 1 move on all $a$ coins.

Compound move 2: $[a,0,0]\rightarrow[0,2^a,0]$, apply type 1 move on 1 of the coin to get $[a-1,2,0]$, then apply compound move 1 to the 2 coins to get $[a-1,0,2^2]$, apply type 2 move and get $[a-2,2^2,0]$, and repeat compound move 1 and type 2 move until $[0,2^a,0]$ is achieve.

Compound move 3: $[a,b,0,0]\rightarrow[a-1,2^b,0,0]$, apply compound move 2 to obtain $[a,0,2^b,0]$ and use type 2 move to get $[a-1,2^b,0,0]$

Compound move 4: $[a,0,0,0]\rightarrow[0,2^{2^{.^{.^{.^2}}}},0,0]$ with $a$ $2$'s. Apply compound move 3 $a$ times.

Compound move 5: $[a,0,0]\rightarrow[b,0,0]$ with $a>b$, use type 2 move $(a-b)$ times.

Let's follow this move: 

$[1,1,1,1,1,1] \rightarrow[0,3,1,1,1,1] \rightarrow[0,2,3,1,1,1]\rightarrow[0,2,1,5,1,1]\rightarrow[0,2,1,1,9,1]\rightarrow[0,2,1,1,0,19]\rightarrow[0,1,19,0,0,0]$


Using Compound move 1, 4 and 5, We can obtain:

$[0,1,19,0,0,0]\rightarrow[0,1,0,X,0,0]\rightarrow[0,0,X,0,0,0]\rightarrow[0,0,0,Y,0,0]\rightarrow[0,0,0,T/4,0,0]\rightarrow[0,0,0,0,T/2,0]\rightarrow[0,0,0,0,0,T]$

, where $X, Y = 2^{2^{.^{.^{.^2}}}}$ where X has $19$ 2's and Y has $X$ 2's, and $Y$ is clearly bigger then $T/4$


See Also

2010 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
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