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Omni-MATH / Let f:X→X, where X={1,2,…,100}, be a function satisfying:
Problem
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problem
Let , where , be a function satisfying:
1) for all ;
2) for any subset of such that , we have .
Find the minimum such that for any such function , there exist a subset of , where , such that .
Plain-text mathematical notation (without MathML)
Let f:X→X, where X={1,2,…,100}, be a function satisfying:
1) f(x)≠x for all x=1,2,…,100;
2) for any subset A of X such that |A|=40, we have A∩f(A)≠∅.
Find the minimum k such that for any such function f, there exist a subset B of X, where |B|=k, such that B∪f(B)=X.Original LaTeX notation
Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:
1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;
2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.
Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such that $B\cup f(B)=X$.Discussion
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