MathDB

1997 IMC

Part of IMC

Subcontests

(6)
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Familiy of finite subsets (sort of set theory)

Suppose FF is a family of finite subsets of N\mathbb{N} and for any 2 sets A,B∈FA,B \in F we have A \cap B \not= \O. (a) Is it true that there is a finite subset YY of N\mathbb{N} such that for any A,B∈FA,B \in F we have A\cap B\cap Y \not= \O? (b) Is the above true if we assume that all members of FF have the same size?

cross the axis uncountably often

Let f:[0,1]→Rf: [0,1]\rightarrow \mathbb{R} continuous. We say that ff crosses the axis at xx if f(x)=0f(x)=0 but ∃y,z∈[x−ϵ,x+ϵ]:f(y)<0<f(z)\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z) for any ϵ\epsilon. (a) Give an example of a function that crosses the axis infinitely often. (b) Can a continuous function cross the axis uncountably often?
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Infinite product

Let α\alpha be a real number, 1<α<21<\alpha<2. (a) Show that α\alpha can uniquely be represented as the infinte product α=(1+1n1)(1+1n2)⋯ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots with nin_i positive integers satisfying ni2≤ni+1n_i^2\le n_{i+1}. (b) Show that α∈Q\alpha\in\mathbb{Q} iff from some kk onwards we have nk+1=nk2n_{k+1}=n_k^2.

space mapping

(a) Let f:Rn×n→Rf: \mathbb{R}^{n\times n}\rightarrow\mathbb{R} be a linear mapping. Prove that ∃!C∈Rn×n\exists ! C\in\mathbb{R}^{n\times n} such that f(A)=Tr(AC),∀A∈Rn×nf(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}. (b) Suppose in addtion that ∀A,B∈Rn×n:f(AB)=f(BA)\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA). Prove that ∃λ∈R:f(A)=λTr(A)\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)
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2
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sum convergence

Let ana_n be a sequence of reals. Suppose ∑an\sum a_n converges. Do these sums converge aswell? (a) a1+a2+(a4+a3)+(a8+...+a5)+(a16+...+a9)+...a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+... (b) a1+a2+(a3)+(a4)+(a5+a7)+(a6+a8)+(a9+a11+a13+a15)+(a10+a12+a14+a16)+(a17+a19+...{a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}

Det of block matrix (trivial)

Let M∈GL2n(K)M \in GL_{2n}(K), represented in block form as M=[ABCD],M−1=[EFGH] M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] Show that det⁡M.det⁡H=det⁡A\det M.\det H=\det A.
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limit of a sequence

Let {ϵn}n=1∞\{\epsilon_n\}^\infty_{n=1} be a sequence of positive reals with lim⁡n→+∞ϵn=0\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0. Find lim⁡n→∞1n∑k=1nln⁡(kn+ϵn) \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right)

Show that g is bounded

Let f∈C3(R)f\in C^3(\mathbb{R}) nonnegative function with f(0)=f′(0)=0,f′′(0)>0f(0)=f'(0)=0, f''(0)>0. Define g(x)g(x) as follows: {g(x)=(f(x)f′(x))′forx≠0g(x)=0forx=0 \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} (a) Show that gg is bounded in some neighbourhood of 00. (b) Is the above true for f∈C2(R)f\in C^2(\mathbb{R})?
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p-norms on integer lattice

For postive integer nn consider the hyperplane R0n=x=(x1x2...xn)∈Rn:∑i=1nxi=0 R_0^n = {x=(x_1x_2...x_n)\in\mathbb{R}^n : \sum\limits^n_{i=1}x_i=0} and the lattice Z0n={y∈R0n: (∀i:yi∈N)} Z_0^n = \{y \in R^n_0 : \ (\forall i: y_i \in \mathbb{N})\} Define the quasi-norm in Rn\mathbb{R}^n by ∥x∥p=∑i=1n∣xi∣pp\|x\|_p= \sqrt[p]{\sum\limits^{n}_{i=1}|x_i|^p} if 0<p<∞0<p<\infty and ∥x∥∞=max⁡i∣xi∣\|x\|_{\infty} = \max\limits_i |x_i|. (a) If x∈R0nx\in R^n_0 so that max⁡xi−min⁡xi≤1\max x_i - \min x_i \le 1 then prove that ∀p∈[1,∞],∀y∈Z0n\forall p \in [1,\infty], \forall y \in Z^n_0 we have ∥x∥p≤∥x+y∥p\|x\|_p\le\|x+y\|_p (b) Prove that for every p∈]0,1[p\in ]0,1[, there exist n∈N,x∈R0n,y∈Z0nn \in \mathbb{N}, x\in R^n_0, y\in Z^n_0 with max⁡xi−min⁡xi≤1\max x_i - \min x_i \le 1 and ∥x∥p>∥x+y∥p\|x\|_p>\|x+y\|_p

cute from IMC

Let XX be an arbitrary set and ff a bijection from XX to XX. Show that there exist bijections g, g′:X→Xg,\ g':X\to X s.t. f=g∘g′, g∘g=g′∘g′=1Xf=g\circ g',\ g\circ g=g'\circ g'=1_X.