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Difference between revisions of "2019 CIME I Problems/Problem 9"

(Created page with "Let <math>\text{N}</math> denote the number of strictly increasing sequences of positive integers <math>a_1,a_2,\cdots, a_{19}</math> satisfying the following two rules<math>:...")
 
 
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Let <math>\text{N}</math> denote the number of strictly increasing sequences of positive integers <math>a_1,a_2,\cdots, a_{19}</math> satisfying the following two rules<math>:</math>
 
Let <math>\text{N}</math> denote the number of strictly increasing sequences of positive integers <math>a_1,a_2,\cdots, a_{19}</math> satisfying the following two rules<math>:</math>
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<math>a_1=1</math> and <math>a_{19}=361,</math>
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* <math>a_1=1</math> and <math>a_{19}=361,</math>
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for any <math>i \neq j,</math> if <math>b_{ij}</math> is the <math>(i \cdot j)^{\text{th}}</math> number not in the sequence<math>,</math> then <math>(a_i-b_{ij})(a_jb_{ij})<0.</math>
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* for any <math>i \neq j,</math> if <math>b_{ij}</math> is the <math>(i \cdot j)^{\text{th}}</math> number not in the sequence<math>,</math> then <math>(a_i-b_{ij})(a_j-b_{ij})<0.</math>
 
Find the largest positive integer <math>k</math> such that <math>2^k</math> divides <math>\text{N}.</math>
 
Find the largest positive integer <math>k</math> such that <math>2^k</math> divides <math>\text{N}.</math>
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=Solution 1=
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{{solution}}
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==See also==
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{{CIME box|year=2019|n=I|num-b=8|num-a=10}}
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[[Category:Intermediate Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 15:28, 6 October 2020

Let $\text{N}$ denote the number of strictly increasing sequences of positive integers $a_1,a_2,\cdots, a_{19}$ satisfying the following two rules$:$

  • $a_1=1$ and $a_{19}=361,$
  • for any $i \neq j,$ if $b_{ij}$ is the $(i \cdot j)^{\text{th}}$ number not in the sequence$,$ then $(a_i-b_{ij})(a_j-b_{ij})<0.$

Find the largest positive integer $k$ such that $2^k$ divides $\text{N}.$

Solution 1

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See also

2019 CIME I (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All CIME Problems and Solutions

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