Difference between revisions of "2014 OIM Problems/Problem 6"

Revision as of 14:22, 14 December 2023

Problem

Given a set $X$ and a function $f: X \to X$ , we say that for each $x \in X$ , $f^1(x)=f(x)$ , and for each $j \ge 1$ , $f^{j+1}=f(f^j(x))$ . We say that $a \in X$ is a fixed point of $f$ if $f(a) = a$ . For each real number $x$ , we define $\pi (x)$ as the number of smaller positive primes less or equal to $x$ . Given a positive integer $n$ , we say that $f : \left\{1, 2, \cdots , n\right\} \to \left\{1, 2, c\dots , n\right\}$ it's "catracha" if $f^{f(k)}(k)=k$ for all $k \in \left\{ 1,2,\cdots,n\right\}$ Prove:

1. If $f$ is catracha, then $f$ has at least $\pi (x)-\pi (\sqrt{x})+1$ fixed points

2. If $n\ge36$ , there exist a catracha function with exactly $\pi (x)-\pi (\sqrt{x})+1$ fixed points

Fun fact irrelevant to the problem: Catracha' is a fried tortilla, covered in fried beans and grated cheese, originating from Honduras.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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@@ Line 6: / Line 6: @@
 . If <math>n\ge36</math>, there exist a ''catracha'' function with exactly <math>\pi (x)-\pi (\sqrt{x})+1</math> fixed points
+Fun fact irrelevant to the problem:  ''Catracha' is a fried tortilla, covered in fried beans and grated cheese, originating from Honduras.
 ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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Difference between revisions of "2014 OIM Problems/Problem 6"

Revision as of 14:22, 14 December 2023

Problem

Solution

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