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International Scientific Olympiad on Mathematics, Iran 2007

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Home » Forum » College Playground » Contests & Research Topics for Undergrads » Other Undergrad Contests

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	dmitin Poincare Conjecture Offline Joined: 05 Feb 2005 Posts: 134 Location: Kyiv, Ukraine		To rate posts you must be logged in #1 International Scientific Olympiad on Mathematics, Iran 2007 olympiad.sanjesh.org The 12th International Scientific Olympiad on Mathematics for University Students 10–13 July 2007 Tehran, Iran I. Mathematical Analysis (Pure and Applied Mathematics) 1. (25 points) Suppose that $f: \mathbb{R}^{n}\to\mathbb{R}^{n}$ is a function satisfying the following two conditions: i) $f(K)$ is compact whenever $K$ is a compact subset of $\mathbb{R}^{n}$ ; ii) $f\left(\bigcap_{n=1}^{\infty}K_{n}\right)=\bigcap_{n=1}^{\infty}f(K_{n})$ whenever $\{K_{n}\}_{n=1}^{\infty}$ is a decreasing sequence of compact subsets of $\mathbb{R}^{n}$ . Prove that $f$ is continuous. 2. (25 points) Suppose $f$ is a real valued twice differentiable function defined on $[a,b]$ . Show that there are $\xi$ , $\eta$ in $[a,b]$ such that $f(\eta)-f(a)\frac{b-\eta}{b-a}-f(b)\frac{\eta-a}{b-a}-\frac{1}{2}(\eta-a)(\eta-b)f''(\xi)=0.$ 3. (25 points) Let $f_{n}: \mathbb{R}\to\mathbb{R}$ be a sequence of functions with the following properties: i) each $f_{n}$ ( $n=1,2,\ldots$ ) is a periodic function with period $T$ ; ii) each $f_{n}$ ( $n=1,2,\ldots$ ) is continuous on $\mathbb{R}$ ; iii) the sequence $\{f_{n}\}_{n=1}^{\infty}$ is uniformly bounded on $\mathbb{R}$ . Then $\{f_{n}\}_{n=1}^{\infty}$ is an equicontinuous sequence of functions on $[0,T]$ . Prove or disprove this statement. 4. (25 points) Suppose $f: [0,\infty)\to\mathbb{R}$ is continuous and $\lim_{x\to\infty}f(x)=L$ . Show that for each $a>0$ $\lim_{n\to+\infty}\int_{0}^{a}f(nx)\,\mathrm{d}x=aL.$ II. Numerical Analysis (Pure and Applied Mathematics) 1. (25 points) Let $g$ be a function defined on $[a,b]$ with properties: i) if $x\in[a,b]$ then $g(x)\in[a,b]$ ; ii) $g'$ is continuous on $[a,b]$ ; iii) for all $x\in[a,b]$ , $\lvert g'(x)\rvert<1$ . 1) Prove that the equation $x=g(x)$ has exactly one root in $[a,b]$ . 2) Show that if condition i) is not satisfied then the equation $x=g(x)$ may have no roots in $[a,b]$ . 2. (25 points) Let $f$ be continuous on $[a,b]$ , $h=\frac{b-a}{n+1}$ and $x_{i+1}-x_{i}=h$ , $i=0,1,\ldots,n$ , $a=x_{0}<x_{1}<\ldots<x_{n}<x_{n+1}=b$ . 1) Find weights of the open Newton-Cotes quadrature rule: $\int_{a}^{b}f(x)\,\mathrm{d}x\approx\sum_{i=1}^{n}w_{i}f(x_{i}).$ 2) Find 3 points open Newton-Cotes formula for $\int_{0}^{4h}f(x)\mathrm{d}x$ . 3) Why open Newton-Cotes quadrature rules are not commonly used? 3. (25 points) Let $\alpha$ be a fixed positive real number and define $x_{n+1}=\frac{x_{n}^{3}+3\alpha x_{n}}{3x_{n}^{2}+\alpha},\quad n=0,1,\ldots$ Assume $x_{0}$ is so that the sequence $\{x_{n}\}$ is convergent. 1) Find $\lim_{n\to\infty}x_{n}$ . 2) Determine the order of convergence of the sequence $\{x_{n}\}$ . 4. (25 points) Find coefficients $a$ , $b$ , $c$ , $d$ , $m$ , $n$ and $p$ in order that for differential equation $y'=f(x,y)$ the Runge-Kutta formulas given as $k_{1}=hf(x,y)$ , $k_{2}=hf(x+mh,y+mk_{1})$ , $k_{3}=hf(x+nh,y+nk_{2})$ , $k_{4}=hf(x+ph,y+pk_{3})$ , $y(x+h)-y(x)\approx ak_{1}+bk_{2}+ck_{3}+dk_{4}$ conform with the Taylor series method of order $h^{4}$ . III. Algebra (Pure Mathematics) 1. (25 points) Let $R$ be a ring with unity. Prove that: 1) If every invertible element is central then every nilpotent element is central. 2) If every nilpotent element is central then every idempotent element is central. 2. (25 points) Let $G$ be a non-cyclic group of order $p^{n}$ where $p$ is a prime number. Prove that $G$ has at least $p+3$ subgroups. 3. (25 points) Let $G$ be a finite group with exactly 50 Sylow 7-subgroups. Let $P\in Syl_{7}(G)$ and $N=N_{G}(P)$ . 1) Prove that $N$ is a maximal subgroup of $G$ . 2) If $N$ has a Sylow 5-subgroup $Q$ and $Q\lhd N$ then prove that $Q\lhd G$ . 4. (25 points) Let $R$ be a ring such that $x^{3}=0$ implies that $x=0$ . Let for each $a,b\in\mathbb{R}$ , $(ab)^{2}=a^{2}b^{2}$ . Prove that $R$ is commutative. Operations Research (Applied Mathematics) 1. (25 points) A matrix $A$ , $m\times n$ , $m\le n$ , with integer components is said to be totally unimodular if every submatrix $B$ composed of a set of $m$ distinct columns of $A$ is so that $\lvert\det(B)\rvert=1$ (every such $B$ is also said to be unimodular). 1) Prove that if an integer matrix $B$ , $m\times m$ , is unimodular then $B^{-1}$ is also an (integer) unimodular matrix. 2) Consider the (LP) problem below: $\min z=c^{T}x$ $\text{s.\,t.\ }Ax=b$ (LP) $x\ge0$ where $A$ is an integer $m\times n$ matrix, $m\le n$ and vector $b$ has integer components. Prove that if $A$ is totally unimodular and (LP) has an optimal solution then the simplex method for solving (LP) will find an optimal integer solution (assuning that (LP) is nondegenerate). 2. (25 points) Consider the linear programming problem below: $\min_{x\in\mathbb{R}^{n}}q^{T}x$ $\text{s.\,t.\ }Mx\ge-q$ (LP) $x\ge0$ where $M=-M^{T}$ and $q$ are given. Prove that: 1) $x\ge0$ is an optimal solution of (LP) if and only if we have $s(x)=Mx+q\ge0,\quad x^{T}s(x)=0.$ 2) Two feasible points $x$ and $y$ for (LP) are optimal points of (LP) if and only if we have $x_{i}s_{i}(y)=y_{i}s_{i}(x),\quad i=1,\ldots,n.$ 3) If $q\ge0$ then the problem (LP) has an optimal solution. 3. (25 points) Prove 1) and 2) using elementary definitions of convex sets and convex functions. 1) Show that the set $S=\{x\in\mathbb{R}^{n}\mid Ax=b,\ x\ge0\}$ is convex. 2) Show that if $x^{}$ is an optimal (local) solution for the problem $\min z=c^{T}x$ $\text{s.\,t.\ }Ax=b$ (LP) $x\ge0$ then $x^{}$ is a global solution. Prove or disprove 3) and 4). 3) Problem (LP) can be infeasible. 4) Problem (LP) can be unbounded. 4. (25 points) Assume that $S$ is a nonempty open convex set in $\mathbb{R}^{n}$ and $f: S\to\mathbb{R}$ is differentiable on $S$ . Prove that if $f$ is convex on $S$ then we have $\bigl(\nabla f(x_{2})-\nabla f(x_{1})\bigr)^{T}(x_{2}-x_{1})\ge0$ for all $x_{1},x_{2}\in S$ . IV. Linear Algebra (Pure and Applied Mathematics) 1. (33.3 points) Let $A\in M_{n}(\mathbb{C})$ and $\mathbb{C}[A]=\{f(A)\mid f(x)\in\mathbb{C}[x]\}$ . Prove that the ring $\mathbb{C}[A]$ has no non-zero nilpotent element if and only if $A$ is diagonalizable. 2. (33.3 points) Prove that if $n$ is a positive integer then there exists a $2\times2$ invertible matrix $A=(a_{ij})$ such that $a_{ij}\in\mathbb{R}$ , $A^{n}=I$ and $A^{k}\ne I$ for every $1\le k<n$ . 3. (33.3 points) Let $X$ be an invertible matrix with columns $X_{1},X_{2},\ldots,X_{n}$ . Let $Y$ be a matrix with columns $X_{2},X_{3},\ldots,X_{n},0$ . Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only 0's for eigenvalues.
			_________________ Cheating is dark side of teaching. Let's vote to vote or not to vote. Let's triple-check everything. lib.mexmat.ru/forum = dxdy.ru Posted: Tue Jul 17, 2007 5:52 pm

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	didilica Yang-Mills Theory Offline Joined: 06 Mar 2006 Posts: 685		To rate posts you must be logged in #2 Here is my solution to the Mathematical Analysis problem 4): Using the substitution $nx=y$ we get that $l=\int_{0}^{a}f(nx)dx=\frac{1}{n}\int_{0}^{na}f(y)dy$ and an application of Cesaro Stoltz lemma shows that $=\int_{na}^{(n+1)a}f(y)dy=af(c_{n}))\rightarrow aL$ , where $c_{n}\in (na, (n+1)a)$ . Note that $\lim f(c_{n})=L$ since $\lim f(x)=L$ .
			_________________ Didi Posted: Tue Jul 17, 2007 9:48 pm

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