목록Study/KAIST POW (13)
AC::MJ LEE
Dividing a circleLet $f$ be a continuous function from $[0,1]$ to a circle. Prove that there exists two closed intervals $I_1 , I_2 \subseteq [0,1]$ such that $I_1 \cap I_2$ has at most one point, $f(I_1)$ and $f(I_2)$ are semicircles, and $f(I_1)\cup f(I_2)$ is a circle. Proof. Let the circle $C$. I will directly construct such intervals. Note that continuous function $f$ preserves connectednes..
Platonic solidsDetermine all platonic solids that can be drawn with the property that all of its vertices are rational points. Proof. For first three platonic solids, it is easy to find specific examples with rational coordinates. Tetrahedron : $(0,0,0) (0,1,1) (1,0,1) (1,1,0)$ Cube : $(0,0,0) (1,0,0) (0,1,0) (0,0,1) (1,1,0) (1,0,1) (0,1,1) (1,1,1)$ Octahedron : $(1,0,0) (-1,0,0) (0,1,0) (0,-1,0..
Rank of a matrix Let $M$ be an $n\times n$ matrix over the reals. Prove that $\operatorname{rank}M = \operatorname{rank}M^2 $ if and only if $\lim_{\lambda\to 0} (\lambda I_n +M)^{-1} M$ exists Proof. Suppose that $\operatorname{rank}M = k$. Then $\exists \{B_1, \ldots ,B_k\}\subset \mathbb{R}^n$ a basis of the $\operatorname{row}M$. Let $B\in \mathbb{R}^{k\times n}$ with $B_1, \ldots ,B_k$ as r..
Product of sines Let $X$ be the set of all positive real number $c$ such that $$\frac{\prod_{k=1}^{n-1}{\sin\frac{k\pi}{2n}}}{c^n}$$ converges as $n$ goes to infinity. Find the infimum of $X$. Proof. Since $\sin\frac{\pi}{2}=1$, we can say $\frac{\prod_{k=1}^{n-1}{\sin\frac{k\pi}{2n}}}{c^n}=\frac{\prod_{k=1}^{n}{\sin\frac{k\pi}{2n}}}{c^n}$ Define $B_n =\frac{\prod_{k=1}^{n-1}{\sin\frac{k\pi}{2n}..
Non-fixed points Let X be a finite non-empty set. Suppose that there is a function $f : X\to X$ such that $f^{20120407}(x)=x$ for all $x\in X$. Prove that the number of elements x in X such that $f(x)\neq x$ is divisible by 20120407 Proof. Since $f^{20120407}(X)=X$, $f(X)=X$. That is, $f$ is surjective. Then $f$ is bijective because X is finite. Define $Y=\{x\in X | f(x)\neq x \}$. Let the numbe..