Difference between revisions of "2014 OIM Problems/Problem 6"
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== Problem == | == Problem == | ||
Given a set <math>X</math> and a function <math>f: X \to X</math>, we say that for each <math>x \in X</math>, <math>f^1(x)=f(x)</math>, and for each <math>j \ge 1</math>, <math>f^{j+1}=f(f^j(x))</math>. We say that <math>a \in X</math> is a fixed point of <math>f</math> if <math>f(a) = a</math>. For each real number <math>x</math>, we define <math>\pi (x)</math> as the number of smaller positive primes less or equal to <math>x</math>. Given a positive integer <math>n</math>, we say that <math>f : \left\{1, 2, \cdots , n\right\} \to \left\{1, 2, c\dots , n\right\}</math> | Given a set <math>X</math> and a function <math>f: X \to X</math>, we say that for each <math>x \in X</math>, <math>f^1(x)=f(x)</math>, and for each <math>j \ge 1</math>, <math>f^{j+1}=f(f^j(x))</math>. We say that <math>a \in X</math> is a fixed point of <math>f</math> if <math>f(a) = a</math>. For each real number <math>x</math>, we define <math>\pi (x)</math> as the number of smaller positive primes less or equal to <math>x</math>. Given a positive integer <math>n</math>, we say that <math>f : \left\{1, 2, \cdots , n\right\} \to \left\{1, 2, c\dots , n\right\}</math> | ||
| − | it's "''catracha''" if <math>f^{f(k)}(k)=k</math> for all <math>k \in \left\{ 1,2,\cdots,n\right\}</math> | + | it's "''catracha''" if <math>f^{f(k)}(k)=k</math> for all <math>k \in \left\{ 1,2,\cdots,n\right\}</math> Prove: |
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| + | 1. If <math>f</math> is ''catracha'', then <math>f</math> has at least <math>\pi (x)-\pi (\sqrt{x})+1</math> fixed points | ||
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| + | 2. If <math>n\ge36</math>, there exist a ''catracha'' function with exactly <math>\pi (x)-\pi (\sqrt{x})+1</math> fixed points | ||
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Revision as of 14:21, 14 December 2023
Problem
Given a set 
and a function 
, we say that for each 
, 
, and for each 
, 
. We say that 
is a fixed point of 
if 
. For each real number 
, we define 
as the number of smaller positive primes less or equal to . Given a positive integer 
, we say that 
it's "catracha" if 
for all 
Prove:
1. If is catracha, then has at least 
fixed points
2. If 
, there exist a catracha function with exactly fixed points
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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